tryalgo package¶

Submodules¶

tryalgo.anagrams module¶

Anagrams christoph dürr - jill-jênn vie - 2013-2019

tryalgo.anagrams.anagrams(S)[source]¶

group a set of words into anagrams

Parameters: S – set of strings
Returns: list of lists of strings
Complexity: $O(n k log k)$ in average, for n words of length at most k. $O(n^2 k log k)$ in worst case due to the usage of a dictionary.

tryalgo.arithm module¶

arithmetic functions christoph dürr - jill-jênn vie - 2013-2019

tryalgo.arithm.bezout(a, b)[source]¶

Bézout coefficients for a and b

Parameters: a,b – non-negative integers
Complexity: O(log a + log b)

tryalgo.arithm.binom(n, k)[source]¶

Binomial coefficients for $n choose k$

Parameters: n,k – non-negative integers
Complexity: O(k)

tryalgo.arithm.binom_modulo(n, k, p)[source]¶

Binomial coefficients for $n choose k$, modulo p

Parameters: n,k – non-negative integers
Complexity: O(k)

tryalgo.arithm.inv(a, p)[source]¶

Inverse of a in ${mathbb Z}_p$

Parameters: a,p – non-negative integers
Complexity: O(log a + log p)

tryalgo.arithm.pgcd(a, b)[source]¶

Greatest common divisor for a and b

Parameters: a,b – non-negative integers
Complexity: O(log a + log b)

tryalgo.arithm_expr_eval module¶

Evaluate an arithmetic expression jill-jenn vie et christoph durr - 2014-2020

IPCELLS http://www.spoj.com/problems/IPCELLS/

tryalgo.arithm_expr_eval.arithm_expr_eval(cell, expr)[source]¶

Evaluates a given expression

Parameters

expr – expression
cell – dictionary variable name -> expression

Returns

numerical value of expression

Complexity

linear

tryalgo.arithm_expr_eval.arithm_expr_parse(line_tokens)[source]¶

Constructs an arithmetic expression tree

Parameters: line_tokens – list of token strings containing the expression
Returns: expression tree
Complexity: linear

tryalgo.arithm_expr_target module¶

Create arithmetic expression approaching target value jill-jênn vie, christoph dürr et jean-christophe filliâtre - 2014-2019

tryalgo.arithm_expr_target.arithm_expr_target(x, target)[source]¶: Create arithmetic expression approaching target value :param x: allowed constants :param target: target value :returns: string in form ‘expression=value’ :complexity: huge

tryalgo.bellman_ford module¶

Single source shortest paths by Bellman-Ford jill-jenn vie et christoph durr - 2014-2018

tryalgo.bellman_ford.bellman_ford(graph, weight, source=0)[source]¶

Single source shortest paths by Bellman-Ford

Parameters

graph – directed graph in listlist or listdict format
weight – can be negative. in matrix format or same listdict graph

Returns

distance table, precedence table, bool

Explanation

bool is True if a negative circuit is reachable from the source, circuits can have length 2.

Complexity

O(|V|*|E|)

tryalgo.bfs module¶

Breadth-first search, bfs and OurQueue christoph dürr - jill-jênn vie - 2015-2019

tryalgo.bfs.bfs(graph, start=0)[source]¶

Shortest path in unweighted graph by BFS

Parameters

graph – directed graph in listlist or listdict format
start (int) – source vertex

Returns

distance table, precedence table

Complexity

O(|V|+|E|)

tryalgo.biconnected_components module¶

bi-connected components, cut vertices and cut cut-nodes jill-jenn vie, christoph durr et louis abraham - 2015-2019

tryalgo.biconnected_components.cut_nodes_edges(graph)[source]¶

Bi-connected components

Parameters: graph – undirected graph. in listlist format. Cannot be in listdict

format. :returns: a tuple with the list of cut-nodes and the list of cut-edges :complexity: O(|V|+|E|)

tryalgo.biconnected_components.cut_nodes_edges2(graph)[source]¶

Bi-connected components, alternative recursive implementation

Parameters: graph – undirected graph. in listlist format. Cannot be in listdict

format. :assumes: graph has about 5000 vertices at most, otherwise memory limit is reached :returns: a tuple with the list of cut-nodes and the list of cut-edges :complexity: O(|V|+|E|) in average, O(|V|+|E|^2) in worst case due to use of dictionary

tryalgo.binary_search module¶

Binary search jill-jênn vie, christoph dürr et louis abraham - 2014-2020

tryalgo.binary_search.continuous_binary_search(f, lo, hi, gap=0.0001)[source]¶

Binary search for a function

Parameters

f – boolean monotone function with f(hi) = True
lo (int) –
hi (int) – with hi >= lo
gap (float) –

Returns

first value x in [lo,hi] such that f(x), x is computed up to some precision

Complexity

O(log((hi-lo)/gap))

tryalgo.binary_search.discrete_binary_search(tab, lo, hi)[source]¶

Binary search in a table

Parameters

tab – boolean monotone table with tab[hi] = True
lo (int) –
hi (int) – with hi >= lo

Returns

first index i in [lo,hi] such that tab[i]

Complexity

O(log(hi-lo))

tryalgo.binary_search.optimized_binary_search(tab, logsize)[source]¶

Binary search in a table using bit operations

Parameters

tab – boolean monotone table of size $2^\textrm{logsize}$ with tab[hi] = True
logsize (int) –

Returns

first i such that tab[i]

Complexity

O(logsize)

tryalgo.binary_search.optimized_binary_search_lower(tab, logsize)[source]¶

Binary search in a table using bit operations

Parameters

tab – boolean monotone table of size $2^\textrm{logsize}$ with tab[0] = False
logsize (int) –

Returns

last i such that not tab[i]

Complexity

O(logsize)

tryalgo.binary_search.ternary_search(f, lo, hi, gap=1e-10)[source]¶

Ternary maximum search for a bitonic function

Parameters: f – boolean bitonic function (increasing then decreasing,

not necessarily strictly) :param int lo: :param int hi: with hi >= lo :param float gap: :returns: value x in [lo,hi] maximizing f(x),

x is computed up to some precision

Complexity: O(log((hi-lo)/gap))

tryalgo.bipartite_matching module¶

Bipartie maximum matching jill-jenn vie et christoph durr - 2014-2018

tryalgo.bipartite_matching.max_bipartite_matching(bigraph)[source]¶

Bipartie maximum matching

Parameters: bigraph – adjacency list, index = vertex in U, value = neighbor list in V
Assumption: U = V = {0, 1, 2, …, n - 1} for n = len(bigraph)
Returns: matching list, match[v] == u iff (u, v) in matching
Complexity: O(|V|*|E|)

tryalgo.bipartite_matching.max_bipartite_matching2(bigraph)[source]¶

Bipartie maximum matching

Parameters: bigraph – adjacency list, index = vertex in U, value = neighbor list in V
Comment: U and V can have different cardinalities
Returns: matching list, match[v] == u iff (u, v) in matching
Complexity: O(|V|*|E|)

tryalgo.bipartite_vertex_cover module¶

Bipartie vertex cover jill-jenn vie et christoph durr - 2014-2018

tryalgo.bipartite_vertex_cover.bipartite_vertex_cover(bigraph)[source]¶

Bipartite minimum vertex cover by Koenig’s theorem

Parameters: bigraph – adjacency list, index = vertex in U, value = neighbor list in V
Assumption: U = V = {0, 1, 2, …, n - 1} for n = len(bigraph)
Returns: boolean table for U, boolean table for V
Comment: selected vertices form a minimum vertex cover, i.e. every edge is adjacent to at least one selected vertex and number of selected vertices is minimum
Complexity: O(|V|*|E|)

tryalgo.closest_points module¶

Closest pair of points trouver la paire de points la plus proche jill-jênn vie, christoph dürr et louis abraham - 2014-2019

tryalgo.closest_points.closest_points(S)[source]¶

Closest pair of points

Parameters: S – list of points
Requires: size of S at least 2
Modifies: changes the order in S
Returns: pair of points p,q from S with minimum Euclidean distance
Complexity: expected linear time

tryalgo.closest_values module¶

Closest values jill-jênn vie et christoph dürr - 2014-2019

tryalgo.closest_values.closest_values(L)[source]¶

Closest values

Parameters: L – list of values
Returns: two values from L with minimal distance
Modifies: the order of L
Complexity: O(n log n), for n=len(L)

tryalgo.convex_hull module¶

Convex hull by Andrew jill-jênn vie et christoph dürr - 2014-2019

tryalgo.convex_hull.andrew(S)[source]¶

Convex hull by Andrew

Parameters: S – list of points as coordinate pairs
Requires: S has at least 2 points
Returns: list of points of the convex hull
Complexity: O(n log n)

tryalgo.dancing_links module¶

Exact set cover by the dancing links algorithm jill-jenn vie et christoph durr - 2014-2018

tryalgo.dancing_links.dancing_links(size_universe, sets)[source]¶

Exact set cover by the dancing links algorithm

Parameters

size_universe – universe = {0, 1, …, size_universe - 1}
sets – list of sets

Returns

list of set indices partitioning the universe, or None

Complexity

huge

tryalgo.dfs module¶

Depth-first search - DFS jill-jênn vie et christoph durr - 2015-2019

tryalgo.dfs.dfs_grid(grid, i, j, mark='X', free='.')[source]¶

DFS on a grid, mark connected component, iterative version

Parameters

grid – matrix, 4-neighborhood
i,j – cell in this matrix, start of DFS exploration
free – symbol for walkable cells
mark – symbol to overwrite visited vertices

Complexity

linear

tryalgo.dfs.dfs_grid_recursive(grid, i, j, mark='X', free='.')[source]¶

DFS on a grid, mark connected component, iterative version

Parameters

grid – matrix, 4-neighborhood
i,j – cell in this matrix, start of DFS exploration
free – symbol for walkable cells
mark – symbol to overwrite visited vertices

Complexity

linear

tryalgo.dfs.dfs_iterative(graph, start, seen)[source]¶

DFS, detect connected component, iterative implementation

Parameters

graph – directed graph in listlist or listdict format
node (int) – to start graph exploration
seen (boolean-table) – will be set true for the connected component containing node.

Complexity

O(|V|+|E|)

tryalgo.dfs.dfs_recursive(graph, node, seen)[source]¶

DFS, detect connected component, recursive implementation

Parameters

graph – directed graph in listlist or listdict format
node (int) – to start graph exploration
seen (boolean-table) – will be set true for the connected component containing node.

Complexity

O(|V|+|E|)

tryalgo.dfs.dfs_tree(graph, start=0)[source]¶

DFS, build DFS tree in unweighted graph

Parameters

graph – directed graph in listlist or listdict format
start (int) – source vertex

Returns

precedence table

Complexity

O(|V|+|E|)

tryalgo.dfs.find_cycle(graph)[source]¶

find a cycle in an undirected graph

Parameters: graph – undirected graph in listlist or listdict format
Returns: list of vertices in a cycle or None
Complexity: O(|V|+|E|)

tryalgo.dijkstra module¶

Shortest paths by Dijkstra jill-jênn vie et christoph dürr - 2015-2018

tryalgo.dijkstra.dijkstra(graph, weight, source=0, target=None)[source]¶

single source shortest paths by Dijkstra

Parameters

graph – directed graph in listlist or listdict format
weight – in matrix format or same listdict graph
source (int) – source vertex
target (int) – if given, stops once distance to target found

Assumes

weights are non-negative

Returns

distance table, precedence table

Complexity

O(|V| + |E|log|V|)

tryalgo.dijkstra.dijkstra_update_heap(graph, weight, source=0, target=None)[source]¶

single source shortest paths by Dijkstra with a heap implementing item updates

Parameters

graph – adjacency list or adjacency dictionary of a directed graph
weight – matrix or adjacency dictionary
source (int) – source vertex
target (int) – if given, stops once distance to target found

Assumes

weights are non-negatif and weights are infinite for non edges

Returns

distance table, precedence table

Complexity

O(|V| + |E|log|V|)

tryalgo.dilworth module¶

Decompose DAG into a minimum number of chains jill-jenn vie et christoph durr - 2015-2018

tryalgo.dilworth.dilworth(graph)[source]¶

Decompose a DAG into a minimum number of chains by Dilworth

Parameters: graph – directed graph in listlist or listdict format
Assumes: graph is acyclic
Returns: table giving for each vertex the number of its chains
Complexity: same as matching

tryalgo.dinic module¶

Maximum flow by Dinic jill-jênn vie et christoph dürr - 2015-2018

tryalgo.dinic.dinic(graph, capacity, source, target)[source]¶

Maximum flow by Dinic

Parameters

graph – directed graph in listlist or listdict format
capacity – in matrix format or same listdict graph
source (int) – vertex
target (int) – vertex

Returns

skew symmetric flow matrix, flow value

Complexity

$O(|V|^2 |E|)$

tryalgo.dist_grid module¶

Distances in a grid jill-jenn vie et christoph durr - 2014-2018 ——————————————–

tryalgo.dist_grid.dist_grid(grid, source, target=None)[source]¶

Distances in a grid by BFS

Parameters

grid – matrix with 4-neighborhood
source ((int,int)) – pair of row, column ind_ices
target ((int,int)) – exploration stops if target is reached

Complexity

linear in grid size

tryalgo.edmonds_karp module¶

Maximum flow by Edmonds-Karp jill-jênn vie et christoph dürr - 2015-2019

tryalgo.edmonds_karp.edmonds_karp(graph, capacity, source, target)[source]¶

Maximum flow by Edmonds-Karp

Parameters

graph – directed graph in listlist or listdict format
capacity – in matrix format or same listdict graph
source (int) – vertex
target (int) – vertex

Returns

flow matrix, flow value

Complexity

$O(|V|*|E|^2)$

tryalgo.eulerian_tour module¶

Eulerian cycle jill-jênn vie et christoph dürr - 2015-2020

tryalgo.eulerian_tour.eulerian_tour_directed(graph)[source]¶

Eulerian tour on a directed graph

Parameters: graph – directed graph in listlist format, cannot be listdict
Assumes: graph is eulerian
Returns: eulerian cycle as a vertex list
Complexity: O(|V|+|E|)

tryalgo.eulerian_tour.eulerian_tour_undirected(graph)[source]¶

Eulerian tour on an undirected graph

Parameters: graph – directed graph in listlist format, cannot be listdict
Assumes: graph is eulerian
Returns: eulerian cycle as a vertex list
Complexity: O(|V|+|E|)

tryalgo.eulerian_tour.is_eulerian_tour(graph, tour)[source]¶

Eulerian tour on an undirected graph

Parameters

graph – directed graph in listlist format, cannot be listdict
tour – vertex list

Returns

test if tour is eulerian

Complexity

O(|V|*|E|) under the assumption that

set membership is in constant time

tryalgo.eulerian_tour.random_eulerien_graph(n)[source]¶

Generates some random eulerian graph

Parameters: n (int) – number of vertices
Returns: undirected graph in listlist representation
Complexity: linear

tryalgo.eulerian_tour.write_cycle(filename, graph, cycle, directed)[source]¶

Write an eulerian tour in DOT format

Parameters

filename – the file to be written in DOT format
graph – graph in listlist format, cannot be listdict
directed (bool) – describes the graph
cycle – tour as a vertex list

Returns

nothing

Complexity

O(|V|^2 + |E|)

tryalgo.fast_exponentiation module¶

Fast Exponentiation jill-jenn vie et christoph durr and louis abraham - 2014-2018

tryalgo.fast_exponentiation.fast_exponentiation(a, b, q)[source]¶

Compute (a pow b) % q, alternative shorter implementation

Parameters

a b (int) – non negative
q (int) – positive

Complexity

O(log b)

tryalgo.fast_exponentiation.fast_exponentiation2(a, b, q)[source]¶

Compute (a pow b) % q

Parameters

a b (int) – non negative
q (int) – positive

Complexity

O(log b)

tryalgo.fenwick module¶

Fenwick tree jill-jenn vie et christoph durr - 2014-2018

class tryalgo.fenwick.Fenwick(t)[source]¶

Bases: object

maintains a tree to allow quick updates and queries

add(a, val)[source]¶

Parameters: a (int) – index in t
Modifies: adds val to t[a]

get(a)[source]¶

Variant, reads t[a]

Parameters: i (int) – negative a will return 0

intervalAdd(a, b, val)[source]¶

Variant, adds val to t[a], to t[a + 1] … and to t[b]

Parameters: a b (int) – with 0 <= a <= b < len(t)

intervalSum(a, b)[source]¶

Parameters: a b (int) – with 0 <= a <= b
Returns: t[a] + … + t[b]

prefixSum(a)[source]¶

Parameters: a (int) – index in t, negative a will return 0
Returns: t[0] + … + t[a]

tryalgo.floyd_warshall module¶

All pairs shortest paths by Floyd-Warshall jill-jênn vie, christoph dürr et pascal ortiz - 2014-2019

tryalgo.floyd_warshall.floyd_warshall(weight)[source]¶

All pairs shortest paths by Floyd-Warshall

Parameters: weight – edge weight matrix
Modifies: weight matrix to contain distances in graph
Returns: True if there are negative cycles
Complexity: $O(|V|^3)$

tryalgo.floyd_warshall.floyd_warshall2(weight)[source]¶

All pairs shortest paths by Floyd-Warshall. An improved implementation by Pascal-Ortiz

Parameters: weight – edge weight matrix
Modifies: weight matrix to contain distances in graph
Returns: True if there are negative cycles
Complexity: $O(|V|^3)$

tryalgo.ford_fulkerson module¶

Maximum flow by Ford-Fulkerson jill-jenn vie et christoph durr - 2014-2018

tryalgo.ford_fulkerson.ford_fulkerson(graph, capacity, s, t)[source]¶

Maximum flow by Ford-Fulkerson

Parameters

graph – directed graph in listlist or listdict format
capacity – in matrix format or same listdict graph
s (int) – source vertex
t (int) – target vertex

Returns

flow matrix, flow value

Complexity

O(|V|*|E|*max_capacity)

tryalgo.freivalds module¶

Test matrix product AB = C by Freivalds jill-jênn vie et christoph dürr - 2015-2020

tryalgo.freivalds.freivalds(A, B, C)[source]¶

Tests matrix product AB=C by Freivalds

Parameters

A – n by n numerical matrix
B – same
C – same

Returns

False with high probability if AB != C

Complexity

$O(n^2)$

tryalgo.gale_shapley module¶

Stable matching by Gale-Shapley jill-jênn vie et christoph durr - 2014-2019

tryalgo.gale_shapley.gale_shapley(men, women)[source]¶

Stable matching by Gale-Shapley

Parameters

men – table of size n, men[i] is preference list of women for men i
women – similar

Returns

matching table, from women to men

Complexity

$O(n^2)$

tryalgo.gauss_jordan module¶

Linear equation system Ax=b by Gauss-Jordan jill-jenn vie et christoph durr - 2014-2018

tryalgo.gauss_jordan.gauss_jordan(A, x, b)[source]¶

Linear equation system Ax=b by Gauss-Jordan

Parameters

A – m by n matrix
x – table of size n
b – table of size m

Modifies

x will contain solution if any

Returns int

0 if no solution, 1 if solution unique, 2 otherwise

Complexity

$O(n^2m)$

tryalgo.graph module¶

Reading graphs from files and writing into files jill-jênn vie et christoph dürr - 2015-2019

class tryalgo.graph.Graph[source]¶

Bases: object

add_arc(name_u, name_v, weight_uv=None)[source]¶

add_edge(name_u, name_v, weight_uv=None)[source]¶

add_node(name)[source]¶

tryalgo.graph.add_reverse_arcs(graph, capac=None)[source]¶

Utility function for flow algorithms that need for every arc (u,v), the existence of an (v,u) arc, by default with zero capacity. graph can be in adjacency list, possibly with capacity matrix capac. or graph can be in adjacency dictionary, then capac parameter is ignored.

Parameters

capac – arc capacity matrix
graph – in listlist representation, or in listdict representation,

in this case capac is ignored :complexity: linear :returns: nothing, but graph is modified

tryalgo.graph.dictdict_to_listdict(dictgraph)[source]¶

Transforms a dict-dict graph representation into a adjacency dictionary representation (list-dict)

Parameters: dictgraph – dictionary mapping vertices to dictionary such that dictgraph[u][v] is weight of arc (u,v)
Complexity: linear
Returns: tuple with graph (listdict), name_to_node (dict),

node_to_name (list)

tryalgo.graph.extract_path(prec, v)[source]¶

extracts a path in form of vertex list from source to vertex v: given a precedence table prec leading to the source

Parameters

prec – precedence table of a tree
v – vertex on the tree

Returns

path from root to v, in form of a list

Complexity

linear

tryalgo.graph.listdict_to_listlist_and_matrix(sparse)[source]¶

Transforms the adjacency list representation of a graph of type listdict into the listlist + weight matrix representation

Parameters: sparse – graph in listdict representation
Returns: couple with listlist representation, and weight matrix
Complexity: linear

tryalgo.graph.listlist_and_matrix_to_listdict(graph, weight=None)[source]¶

Transforms the weighted adjacency list representation of a graph of type listlist + optional weight matrix into the listdict representation

Parameters

graph – in listlist representation
weight – optional weight matrix

Returns

graph in listdict representation

Complexity

linear

tryalgo.graph.make_flow_labels(graph, flow, capac)[source]¶

Generate arc labels for a flow in a graph with capacities.

Parameters

graph – adjacency list or adjacency dictionary
flow – flow matrix or adjacency dictionary
capac – capacity matrix or adjacency dictionary

Returns

listdic graph representation, with the arc label strings

tryalgo.graph.matrix_to_listlist(weight)[source]¶

transforms a squared weight matrix in a adjacency table of type listlist encoding the directed graph corresponding to the entries of the matrix different from None

Parameters: weight – squared weight matrix, weight[u][v] != None iff arc (u, v)

exists :complexity: linear :returns: the unweighted directed graph in the listlist representation,

listlist[u] contains all v for which arc (u,v) exists.

tryalgo.graph.read_graph(filename, directed=False, weighted=False, default_weight=None)[source]¶

Read a graph from a text file

Parameters

filename – plain text file. All numbers are separated by space. Starts with a line containing n (#vertices) and m (#edges). Then m lines follow, for each edge. Vertices are numbered from 0 to n-1. Line for unweighted edge u,v contains two integers u, v. Line for weighted edge u,v contains three integers u, v, w[u,v].
directed – true for a directed graph, false for undirected
weighted – true for an edge weighted graph

Returns

graph in listlist format, possibly followed by weight matrix

Complexity

O(n + m) for unweighted graph, $O(n^2)$ for weighted graph

tryalgo.graph.readtab(fi, ty)[source]¶

Reads a line from file with a space separated list: of items of type ty

Parameters

file – input stream, for example sys.stdin
ty – a type, for example int

Returns

a tuple with elements of type ty

tryalgo.graph.readval(file, ty)[source]¶

Reads a line from file with an item of type ty

Parameters

file – input stream, for example sys.stdin
ty – a type, for example int

Returns

an element of type ty

tryalgo.graph.tree_adj_to_prec(graph, root=0)[source]¶

Transforms a tree given as adjacency list into predecessor table form. if graph is not a tree: will return a DFS spanning tree

Parameters: graph – directed graph in listlist or listdict format
Returns: tree in predecessor table representation
Complexity: linear

tryalgo.graph.tree_prec_to_adj(prec, root=0)[source]¶

Transforms a tree given as predecessor table into adjacency list form

Parameters: prec – predecessor table representing a tree, prec[u] == v iff u is

successor of v,: except for the root where prec[root] == root

Parameters: root – root vertex of the tree
Returns: undirected graph in listlist representation
Complexity: linear

tryalgo.graph.write_graph(dotfile, graph, directed=False, node_label=None, arc_label=None, comment='', node_mark={}, arc_mark={})[source]¶

Writes a graph to a file in the DOT format

Parameters

dotfile – the filename.
graph – directed graph in listlist or listdict format
directed – true if graph is directed, false if undirected
weight – in matrix format or same listdict graph or None
node_label – vertex label table or None
arc_label – arc label matrix or None
comment – comment string for the dot file or None
node_mark – set of nodes to be shown in gray
arc_marc – set of arcs to be shown in red

Complexity

O(|V| + |E|)

tryalgo.graph01 module¶

Shortest path in a 0,1 weighted graph jill-jenn vie et christoph durr - 2014-2018

tryalgo.graph01.dist01(graph, weight, source=0, target=None)[source]¶

Shortest path in a 0,1 weighted graph

Parameters

graph – directed graph in listlist or listdict format
weight – matrix or adjacency dictionary
source (int) – vertex
target – exploration stops once distance to target is found

Returns

distance table, predecessor table

Complexity

O(|V|+|E|)

tryalgo.horn_sat module¶

Solving Horn SAT

christoph dürr - 2016-2020

clauses are numbered starting from 0 variables are strings (identifier)

solution : set of variables that are set to true posvar_in_clause : maps clause to the unique positive variable in clause (or None) clause_with_negvar : maps variable v to all clauses that contain not(v)

every clause has a score: number of its negative variables that are not in solution sol pool : maps score to clauses of that score

tryalgo.horn_sat.horn_sat(formula)[source]¶

Solving a HORN Sat formula

Parameters: formula – list of couple(posvar, negvars). negvars is a list of the negative variables (can be empty) posvar is the positive variable (can be None) Variables can be any hashable objects: integers, strings…
Returns: None if formula is not satisfiable, else a minimal set of vars that have to be set to true in order to satisfy the formula.
Complexity: linear

tryalgo.horn_sat.read(filename)[source]¶

reads a Horn SAT formula from a text file

File format: # comment A # clause with unique positive literal :- A # clause with unique negative literal A :- B, C, D # clause where A is positive and B,C,D negative # variables are strings without spaces

tryalgo.huffman module¶

Huffman code jill-jenn vie et christoph durr - 2014-2019

tryalgo.huffman.extract(code, tree, prefix)[source]¶

Extract Huffman code from a Huffman tree

Parameters

code – a dictionary that will contain the constructed code. should initially be empty.
tree – a node of the tree. Leafs are of the form (frequency, symbol). Inner nodes are of the form [left_sub_tree, right_sub_tree].
prefix – a list with the 01 characters encoding the path from the root to the node tree

Complexity

O(n), where n = number of nodes in tree

tryalgo.huffman.huffman(freq)[source]¶

Huffman code

Parameters: freq – dictionary with frequencies for each item
Returns: dictionary with binary code string for each item
Complexity: O(n log n), where n = len(freq)

tryalgo.interval_cover module¶

Minimum interval cover jill-jênn vie et christoph dürr - 2014-2020

tryalgo.interval_cover.interval_cover(I)[source]¶

Minimum interval cover

Parameters: I – list of closed intervals
Returns: minimum list of points covering all intervals
Complexity: O(n log n)

tryalgo.interval_tree module¶

Interval tree christoph dürr - jill-jênn vie - 2013-2018

tryalgo.interval_tree.interval_tree(intervals)[source]¶

Construct an interval tree

Parameters: intervals – list of half-open intervals encoded as value pairs [left, right)
Assumes: intervals are lexicographically ordered >>> assert intervals == sorted(intervals)
Returns: the root of the interval tree
Complexity: $O(n)$

tryalgo.interval_tree.intervals_containing(t, p)[source]¶

Query the interval tree

Parameters

t – root of the interval tree
p – value

Returns

a list of intervals containing p

Complexity

O(log n + m), where n is the number of intervals in t, and m the length of the returned list

tryalgo.intervals_union module¶

Union of intervals jill-jênn vie et christoph dürr - 2014-2019

tryalgo.intervals_union.intervals_union(S)[source]¶

Union of intervals

Parameters: S – list of pairs (low, high) defining intervals [low, high)
Returns: ordered list of disjoint intervals with the same union as S
Complexity: O(n log n)

tryalgo.knapsack module¶

Knapsack jill-jênn vie et christoph dürr - 2015-2019

tryalgo.knapsack.knapsack(p, v, cmax)[source]¶

Knapsack problem: select maximum value set of items if total size not more than capacity

Parameters

p – table with size of items
v – table with value of items
cmax – capacity of bag

Requires

number of items non-zero

Returns

value optimal solution, list of item indexes in solution

Complexity

O(n * cmax), for n = number of items

tryalgo.knapsack.knapsack2(p, v, cmax)[source]¶

Knapsack problem: select maximum value set of items if total size not more than capacity. alternative implementation with same behavior.

Parameters

p – table with size of items
v – table with value of items
cmax – capacity of bag

Requires

number of items non-zero

Returns

value optimal solution, list of item indexes in solution

Complexity

O(n * cmax), for n = number of items

tryalgo.knuth_morris_pratt module¶

Find the length of maximal borders by Knuth-Morris-Pratt

jill-jênn vie, christoph dürr et louis abraham - 2014-2019 inspired from a practical lesson (TP) from Yves Lemaire

tryalgo.knuth_morris_pratt.knuth_morris_pratt(s, t)[source]¶

Find a substring by Knuth-Morris-Pratt

Parameters

s – the haystack string
t – the needle string

Returns

index i such that s[i: i + len(t)] == t, or -1

Complexity

O(len(s) + len(t))

tryalgo.knuth_morris_pratt.maximum_border_length(w)[source]¶

Maximum string borders by Knuth-Morris-Pratt

Parameters: w – string
Returns: table f such that f[i] is the longest border length of w[:i + 1]
Complexity: linear

tryalgo.knuth_morris_pratt.powerstring_by_border(u)[source]¶

Power string by Knuth-Morris-Pratt

Parameters: x – string
Returns: largest k such that there is a string y with x = y^k
Complexity: O(len(x))

tryalgo.knuth_morris_pratt.powerstring_by_find(u)[source]¶

Power string using the python find method

Parameters: x – string
Returns: largest k such that there is a string y with x = y^k
Complexity: O(len(x))

tryalgo.kruskal module¶

Minimum spanning tree by kruskal jill-jenn vie et christoph durr - 2014-2018

class tryalgo.kruskal.UnionFind(n)[source]¶

Bases: object

Maintains a partition of {0, …, n-1}

find(x_index)[source]¶

Returns: identifier of part containing x_index
Complex_indexity: O(inverse_ackerman(n))

union(x_index, y_index)[source]¶: Merges part that contain x and part containing y :returns: False if x_index, y_index are already in same part :complexity: O(inverse_ackerman(n))

tryalgo.kruskal.dist(a, b)[source]¶: distance between point a and point b

tryalgo.kruskal.kruskal(graph, weight)[source]¶

Minimum spanning tree by Kruskal

Parameters

graph – undirected graph in listlist or listdict format
weight – in matrix format or same listdict graph

Returns

list of edges of the tree

Complexity

O(|E|log|E|)

tryalgo.kuhn_munkres module¶

Maximum profit bipartite matching by Kuhn-Munkres jill-jenn vie, christoph durr and samuel tardieu - 2014-2019

primal LP

max sum_{u,v} w[u,v] * x[u,v]

such that for all u in U: sum_v x[u,v] == 1 (l[u])

for all v in V: sum_u x[u,v] <= 1 (l[v])

for all u,v: x[u,v] >= 0

dual LP

min sum_u l[u] + sum_v l[v]

such that for all u,v: l[u] + l[v] >= w[u,v] (*)

for all u in U: l[u] is arbitrary (free variable) for all v in V: l[v] >= 0

primal-dual algorithm:

Start with trivial solution l for dual and with trivial non-solution x for primal.

Iteratively improve primal or dual solution, maintaining complementary slackness conditions.

tryalgo.kuhn_munkres.kuhn_munkres(G, TOLERANCE=1e-06)[source]¶

Maximum profit bipartite matching by Kuhn-Munkres

Parameters

G – weight matrix where G[u][v] is the weight of edge (u,v),
TOLERANCE – a value with absolute value below tolerance is considered as being zero. If G consists of integer or fractional values then TOLERANCE can be chosen 0.

Requires

graph (U,V,E) is complete bi-partite graph with len(U) <= len(V) float(‘-inf’) or float(‘inf’) entries in G are allowed but not None.

Returns

matching table from U to V, value of matching

Complexity

$O(|U|^2 |V|)$

tryalgo.kuhn_munkres_n4 module¶

kuhn_munkres_n4 jill-jênn vie et christoph dürr - 2014-2019

tryalgo.kuhn_munkres_n4.kuhn_munkres(G)[source]¶

Maximum profit perfect matching

for minimum cost perfect matching just inverse the weights

Parameters: G – squared weight matrix of a complete bipartite graph
Complexity: $O(n^4)$

tryalgo.laser_mirrors module¶

Orienting mirrors to allow connectivity by a laser beam jill-jênn vie et christoph dürr - 2014-2019

tryalgo.laser_mirrors.laser_mirrors(rows, cols, mir)[source]¶

Orienting mirrors to allow reachability by laser beam

Parameters

rows (int) –
cols (int) – rows and cols are the dimension of the grid
mir – list of mirror coordinates, except mir[0]= laser entrance, mir[-1]= laser exit.

Complexity

$O(2^n)$

tryalgo.laser_mirrors.solve(succ, orien, i, direc)[source]¶

Can a laser leaving mirror i in direction direc reach exit ?

Parameters

i – mirror index
direc – direction leaving mirror i
orient – orient[i]=orientation of mirror i
succ – succ[i][direc]=succ mirror reached when leaving i in direction direc

tryalgo.left_right_inversions module¶

Left and right inversions in a table christoph dürr - 2016-2019

tryalgo.left_right_inversions.left_right_inversions(tab)[source]¶

Compute left and right inversions of each element of a table.

Parameters: tab – list with comparable elements
Returns: lists left and right. left[j] = the number of

i<j such that tab[i] > tab[j].: right[i] = the number of i<j such that tab[i] > tab[j].

Complexity: O(n log n)

tryalgo.levenshtein module¶

Levenshtein edit distance jill-jenn vie et christoph durr - 2014-2018

tryalgo.levenshtein.levenshtein(x, y)[source]¶

Levenshtein edit distance

Parameters

x –
y – strings

Returns

distance

Complexity

O(|x|*|y|)

tryalgo.longest_common_subsequence module¶

Longest increasing subsequence jill-jênn vie et christoph dürr - 2014-2019

tryalgo.longest_common_subsequence.longest_common_subsequence(x, y)[source]¶

Longest common subsequence

Dynamic programming

Parameters

x –
y – x, y are lists or strings

Returns

longest common subsequence in form of a string

Complexity

O(|x|*|y|)

tryalgo.longest_increasing_subsequence module¶

Longest increasing subsequence jill-jenn vie et christoph durr - 2014-2018

tryalgo.longest_increasing_subsequence.longest_increasing_subsequence(x)[source]¶

Longest increasing subsequence

Parameters: x – sequence
Returns: longest strictly increasing subsequence y
Complexity: O(|x|*log(|y|))

tryalgo.lowest_common_ancestor module¶

Lowest common ancestor jill-jenn vie et christoph durr - 2014-2018 http://leetcode.com/2011/11/longest-palindromic-substring-part-ii.html

class tryalgo.lowest_common_ancestor.LowestCommonAncestorRMQ(graph)[source]¶

Bases: object

Lowest common ancestor data structure using a reduction to range minimum query

query(u, v)[source]¶

Returns: the lowest common ancestor of u and v
Complexity: O(log n)

class tryalgo.lowest_common_ancestor.LowestCommonAncestorShortcuts(prec)[source]¶

Bases: object

Lowest common ancestor data structure using shortcuts to ancestors

query(u, v)[source]¶

Returns: the lowest common ancestor of u and v
Complexity: O(log n)

tryalgo.lowest_common_ancestor.log2ceil(n)[source]¶: log of n in base 2 rounded up

tryalgo.lowest_common_ancestor.log2floor(n)[source]¶: log of n in base 2 rounded down

tryalgo.majority module¶

Majority jill-jenn vie et christoph durr - 2014-2019

tryalgo.majority.majority(L)[source]¶

Majority

Parameters: L – list of elements
Returns: element that appears most in L, tie breaking with smallest element
Complexity: $O(nk)$ in average, where n = len(L) and k = max(w for w in L) $O(n^2k)$ in worst case due to the use of a dictionary

tryalgo.manacher module¶

Longest palindrome in a string by Manacher jill-jênn vie et christoph dürr - 2014-2019

http://leetcode.com/2011/11/longest-palindromic-substring-part-ii.html

Algorithme de Manacher problème: plus long palindrome entrée: chaîne s sortie: indices i, j tel que s[i:j] est un palindrome

et que j-i est maximal et i maximal

complexité: temps linéaire

tous les indices réfèrent à une chaîne fictive t de la forme “^#a#b#a#a#$” si s=”abaa” invariant: pour chaque préfixe vu on maintient un palindrome centré en c et de bord droit r qui maximise r ainsi que p[i] = plus grand rayon d’un palindrome centré en i

tryalgo.manacher.manacher(s)[source]¶

Longest palindrome in a string by Manacher

Parameters: s – string
Requires: s is not empty
Returns: i,j such that s[i:j] is the longest palindrome in s
Complexity: O(len(s))

tryalgo.matrix_chain_mult module¶

Matrix chain multiplication multiplication de matrices jill-jenn vie et christoph durr - 2014-2018

tryalgo.matrix_chain_mult.matrix_chain_mult(M)[source]¶

Matrix chain multiplication

Parameters: M – list of matrices
Returns: M[0] * … * M[-1], computed in time optimal order
Complexity: whatever is needed by the multiplications

tryalgo.matrix_chain_mult.matrix_mult_opt_order(M)[source]¶

Matrix chain multiplication optimal order

Parameters: M – list of matrices
Returns: matrices opt, arg, such that opt[i][j] is the optimal number of operations to compute M[i] * … * M[j] when done in the order (M[i] * … * M[k]) * (M[k + 1] * … * M[j]) for k = arg[i][j]
Complexity: $O(n^2)$

tryalgo.max_interval_intersec module¶

Sweepline algorithm technique jill-jênn vie et christoph dürr - 2014-2019

tryalgo.max_interval_intersec.max_interval_intersec(S)[source]¶

determine a value that is contained in a largest number of given intervals

Parameters: S – list of half open intervals
Complexity: O(n log n), where n = len(S)

tryalgo.merge_ordered_lists module¶

Merge two ordered lists jill-jênn vie et christoph dürr - 2014-2019

tryalgo.merge_ordered_lists.merge(x, y)[source]¶

Merge two ordered lists

Parameters

x –
y – x, y are nondecreasing ordered lists

Returns

union of x and y in order

Complexity

linear

tryalgo.min_mean_cycle module¶

Minimum mean cycle by Karp jill-jenn vie et christoph durr - 2014-2018

tryalgo.min_mean_cycle.min_mean_cycle(graph, weight, start=0)[source]¶

Minimum mean cycle by Karp

Parameters

graph – directed graph in listlist or listdict format
weight – in matrix format or same listdict graph
start (int) – vertex that should be contained in cycle

Returns

cycle as vertex list, average arc weights or None if there is no cycle from start

Complexity

O(|V|*|E|)

tryalgo.next_permutation module¶

Next permutation prochaine permuation jill-jênn vie et christoph dürr - 2014-2020

tryalgo.next_permutation.convert(word, ass)[source]¶: solves a cryptogram in the style SEND + MORE = MONEY

tryalgo.next_permutation.next_permutation(tab)[source]¶

find the next permutation of tab in the lexicographical order

Parameters: tab – table with n elements from an ordered set
Modifies: table to next permutation
Returns: False if permutation is already lexicographical maximal
Complexity: O(n)

tryalgo.next_permutation.solve_word_addition(S)[source]¶: returns number of solutions

tryalgo.our_heap module¶

A min heap christoph dürr et jill-jênn vie - 2015-2019

class tryalgo.our_heap.OurHeap(items)[source]¶

Bases: object

min heap

heap: is the actual heap, heap[1] = index of the smallest element
rank: inverse of heap with rank[x]=i iff heap[i]=x
n: size of the heap

Complexity: init O(n log n), len O(1), other operations O(log n) in expectation and O(n) in worst case, due to the usage of a dictionary

down(i)[source]¶: the value of heap[i] has increased. Maintain heap invariant.

pop()[source]¶: Remove and return smallest element

push(x)[source]¶: Insert new element x in the heap. Assumption: x is not already in the heap

up(i)[source]¶: The value of heap[i] has decreased. Maintain heap invariant.

update(old, new)[source]¶: Replace an element in the heap

tryalgo.our_queue module¶

A FIFO queue christoph dürr - jill-jênn vie - 2015-2019

class tryalgo.our_queue.OurQueue[source]¶

Bases: object

A FIFO queue

Complexity:: all operators in amortized constant time, except __str__ which is linear

pop()[source]¶

push(obj)[source]¶

tryalgo.our_std module¶

Our Standards Jill-Jênn Vie et Christoph Dürr - 2020

tryalgo.our_std.readarray(typ)[source]¶: function to read an array

tryalgo.our_std.readint()[source]¶: function to read an integer from stdin

tryalgo.our_std.readmatrix(n)[source]¶: function to read a matrix

tryalgo.our_std.readstr()[source]¶: function to read a string from stdin

tryalgo.partition_refinement module¶

Partition refinement christoph dürr - 2016-2019

log: 10/11/2016 modified to preserve class order after refinement: 15/11/2016 this was nonsense, moved back

class tryalgo.partition_refinement.PartitionRefinement(n)[source]¶

Bases: object

This data structure implements an order preserving partition with refinements.

order()[source]¶: Produce a flatten list of the partition, ordered by classes

refine(pivot)[source]¶: Split every class C in the partition into C intersection pivot and C setminus pivot complexity: linear in size of pivot

tolist()[source]¶: produce a list representation of the partition

tryalgo.permutation_rank module¶

Permutation rank christoph dürr - 2016-2019

tryalgo.permutation_rank.permutation_rank(p)[source]¶

Given a permutation of {0,..,n-1} find its rank according to lexicographical order

param p

list of length n containing all integers from 0 to n-1

returns

rank between 0 and n! -1

beware

computation with big numbers

complexity

O(n^2)

tryalgo.permutation_rank.rank_permutation(r, n)[source]¶

Given r and n find the permutation of {0,..,n-1} with rank according to lexicographical order equal to r

param r n

integers with 0 ≤ r < n!

returns

permutation p as a list of n integers

beware

computation with big numbers

complexity

O(n^2)

tryalgo.polygon module¶

Area of polygone mesures polygone jill-jenn vie et christoph durr - 2014-2018

tryalgo.polygon.area(p)[source]¶

Area of a polygone

Parameters: p – list of the points taken in any orientation, p[0] can differ from p[-1]
Returns: area
Complexity: linear

tryalgo.polygon.is_simple(polygon)[source]¶

Test if a rectilinear polygon is is_simple

Parameters: polygon – list of points as (x,y) pairs along the closed polygon
Returns: True if the segements do not intersect
Complexity: O(n log n) for n=len(polygon)

tryalgo.predictive_text module¶

Predictive text for mobile phones jill-jenn vie et christoph durr and louis abraham - 2014-2019

tryalgo.predictive_text.predictive_text(dic)[source]¶

Predictive text for mobile phones

Parameters: dic – associates weights to words from [a-z]*
Returns: a dictionary associating to words from [2-9]* a corresponding word from the dictionary with highest weight
Complexity: linear in total word length

tryalgo.predictive_text.propose(prop, seq)[source]¶: wrapper to access a dictionary even for non-present keys

tryalgo.primes module¶

Prime numbers by Eratosthene nombre premiers <n jill-jenn vie et christoph durr - 2014-2018

tryalgo.primes.eratosthene(n)[source]¶

Prime numbers by sieve of Eratosthene

Parameters: n – positive integer
Assumes: n > 2
Returns: list of prime numbers <n
Complexity: O(n loglog n)

tryalgo.primes.gries_misra(n)[source]¶

Prime numbers by the sieve of Gries-Misra Computes both the list of all prime numbers less than n, and a table mapping every integer 2 ≤ x < n to its smallest prime factor

Parameters: n – positive integer
Returns: list of prime numbers, and list of prime factors
Complexity: O(n)

tryalgo.rabin_karp module¶

Find substrings by Rabin-Karp jill-jenn vie et christoph durr - 2015-2019

tryalgo.rabin_karp.matches(s, t, i, j, k)[source]¶: Checks whether s[i:i + k] is equal to t[j:j + k]. We used a loop to ease the implementation in other languages.

tryalgo.rabin_karp.rabin_karp_factor(s, t, k)[source]¶

Find a common factor by Rabin-Karp

Parameters

s (string) – haystack
t (string) – needle
k (int) – factor length

Returns

(i, j) such that s[i:i + k] == t[j:j + k] or None. In case of tie, lexicographical minimum (i, j) is returned

Complexity

O(len(s) + len(t)) in expected time, and O(len(s) + len(t) * k) in worst case

tryalgo.rabin_karp.rabin_karp_matching(s, t)[source]¶

Find a substring by Rabin-Karp

Parameters

s – the haystack string
t – the needle string

Returns

index i such that s[i: i + len(t)] == t, or -1

Complexity

O(len(s) + len(t)) in expected time, and O(len(s) * len(t)) in worst case

tryalgo.rabin_karp.roll_hash(old_val, out_digit, in_digit, last_pos)[source]¶

tryalgo.range_minimum_query module¶

Range minimum query Minimum d’une plage — range minimum query jill-jenn vie et christoph durr - 2014-2019

class tryalgo.range_minimum_query.LazySegmentTree(tab)[source]¶

Bases: object

maintains a tree to allow quick updates and queries on a table.

This is more general than a Fenwick tree or a tree for MinRangeQuery. Here queries and updates act on index ranges. Updates can be set a range to a value or add a value to a range. Queries can be max, min and sum over an index range. All operations run in time O(log n) for a the table size n. The given ranges are in the form [i,j] where i is included and j excluded. In the recursive calls, node is the index of a node in the tree, and left, right its range. Values can be any numerical values allowing max, min, and sum, such as integers, floating point numbers or fractions (from the class Fraction). Updates over an empty range is valid and does nothing. Queries over an empty range is valid and returns the neutral value -inf, +inf or 0.

If the node is cleared, then maxval, minval, sumval represent for each node the query responses over the corresponding index ranges. If the node is not clean, it means that lazyset and/or lazyadd contain suspendet update instructions for that node. Clearing a node means propagating these values to the descents in the subtrees, and updating maxval,minval and sumval for that node.

add(i, j, val)[source]¶

max(i, j)[source]¶

min(i, j)[source]¶

set(i, j, val)[source]¶

sum(i, j)[source]¶

class tryalgo.range_minimum_query.RangeMinQuery(t, INF=inf)[source]¶

Bases: object

Range minimum query

maintains a table t, can read/write items t[i], and query range_min(i,k) = min{ t[i], t[i + 1], …, t[k - 1]} :complexity: all operations in O(log n), for n = len(t)

range_min(i, k)[source]¶

Returns: min{ t[i], t[i + 1], …, t[k - 1]}
Complexity: O(log len(t))

tryalgo.rectangles_from_grid module¶

Largest area rectangle in a binary matrix plus grand rectangle monochromatique jill-jenn vie et christoph durr - 2014-2018

tryalgo.rectangles_from_grid.rectangles_from_grid(P, black=1)[source]¶

Largest area rectangle in a binary matrix

Parameters

P – matrix
black – search for rectangles filled with value black

Returns

area, left, top, right, bottom of optimal rectangle consisting of all (i,j) with left <= j < right and top <= i <= bottom

Complexity

linear

tryalgo.rectangles_from_histogram module¶

Largest Rectangular Area in a Histogram jill-jenn vie et christoph durr - 2014-2018

tryalgo.rectangles_from_histogram.rectangles_from_histogram(H)[source]¶

Largest Rectangular Area in a Histogram

Parameters: H – histogram table
Returns: area, left, height, right, rect. is [0, height] * [left, right)
Complexity: linear

tryalgo.rectangles_from_points module¶

How many rectangles can be formed from a set of points jill-jenn vie et christoph durr - 2014-2018

tryalgo.rectangles_from_points.rectangles_from_points(S)[source]¶

How many rectangles can be formed from a set of points

Parameters: S – list of points, as coordinate pairs
Returns: the number of rectangles
Complexity: $O(n^2)$

tryalgo.roman_numbers module¶

Evaluate an arithmetic expression jill-jênn vie et christoph dürr - 2014-2019

tryalgo.roman_numbers.int2roman(val)[source]¶

Code roman number

Parameters: val – integer between 1 and 9999
Returns: the corresponding roman number
Complexity: linear (if that makes sense for constant bounded input size)

tryalgo.roman_numbers.roman2int(s)[source]¶

Decode roman number

Parameters: s – string representing a roman number between 1 and 9999
Returns: the decoded roman number
Complexity: linear (if that makes sense for constant bounded input size)

tryalgo.scalar module¶

Permute vector to minimize scalar product jill-jênn vie et christoph dürr - 2014-2019

tryalgo.scalar.min_scalar_prod(x, y)[source]¶

Permute vector to minimize scalar product

Parameters

x –
y – x, y are vectors of same size

Returns

min sum x[i] * y[sigma[i]] over all permutations sigma

Complexity

O(n log n)

tryalgo.shortest_cycle module¶

christoph durr, finn voelkel and louis abraham - 2016-2019

Find shortest simple cycle O(V*E) footnote (1) here you can add parity check of cycle_len if only even cycles are requested

tryalgo.shortest_cycle.powergraph(graph, k)[source]¶

Compute the k-th: powergraph which has an edge u,v for every vertex pair of distance at most k in the given graph.

Parameters

graph – undirected graph in listlist or listdict format
k – non-negative integer.

Complexity

O(V^3)

tryalgo.shortest_cycle.shortest_cycle(graph)[source]¶

Finding a shortest cycle in an undirected unweighted graph

Parameters: graph – undirected graph in listlist or listdict format
Returns: None or a list C describing a shortest cycle
Complexity: O(|V|*|E|)

tryalgo.skip_list module¶

skip-list louis abraham - 2017-2019

Inspired by https://kunigami.blog/2012/09/25/skip-lists-in-python/ count contains the gap between the positions (https://www.cs.bgu.ac.il/~ds112/wiki.files/ds112_ps7.pdf)

class tryalgo.skip_list.AbstractSkipList[source]¶

Bases: object

find(key, update=None)[source]¶

getkth(k)[source]¶: starts from 0

insert(node)[source]¶

lastKey(key)[source]¶: lastKey(key) < key

nextKey(key)[source]¶: nextKey(key) >= key

nextNode(key, update=None)[source]¶

static randomHeight()[source]¶

remove(key)[source]¶

class tryalgo.skip_list.SortedDict(iterable=())[source]¶

Bases: tryalgo.skip_list.AbstractSkipList

class Node(key, val, next, count)¶

Bases: tuple

count¶: Alias for field number 3

key¶: Alias for field number 0

next¶: Alias for field number 2

val¶: Alias for field number 1

keys()[source]¶

class tryalgo.skip_list.SortedSet(iterable=())[source]¶

Bases: tryalgo.skip_list.AbstractSkipList

class Node(key, next, count)¶

Bases: tuple

count¶: Alias for field number 2

key¶: Alias for field number 0

next¶: Alias for field number 1

add(key)[source]¶

pop()[source]¶: Pops the first element

tryalgo.skip_list.random() → x in the interval [0, 1).¶

tryalgo.strongly_connected_components module¶

Strongly connected components composantes fortement connexes jill-jênn vie et christoph dürr - 2015-2018

tryalgo.strongly_connected_components.kosaraju(graph)[source]¶

Strongly connected components by Kosaraju

Parameters: graph – directed graph in listlist format, cannot be listdict
Returns: list of lists for each component
Complexity: linear

tryalgo.strongly_connected_components.reverse(graph)[source]¶: replace all arcs (u, v) by arcs (v, u) in a graph

tryalgo.strongly_connected_components.tarjan(graph)[source]¶

Strongly connected components by Tarjan, iterative implementation

Parameters: graph – directed graph in listlist format, cannot be listdict
Returns: list of lists for each component
Complexity: linear

tryalgo.strongly_connected_components.tarjan_recursif(graph)[source]¶

Strongly connected components by Tarjan, recursive implementation

Parameters: graph – directed graph in listlist format, cannot be listdict
Returns: list of lists for each component
Complexity: linear

tryalgo.subsetsum module¶

subsetsum jill-jenn vie et christoph durr - 2015-2018

tryalgo.subsetsum.coin_change(x, R)[source]¶

Coin change

Parameters

x – table of non negative values
R – target value

Returns bool

True if there is a non negative linear combination

of x that has value R :complexity: O(n*R)

tryalgo.subsetsum.subset_sum(x, R)[source]¶

Subsetsum

Parameters

x – table of non negative values
R – target value

Returns bool

True if a subset of x sums to R

Complexity

O(n*R)

tryalgo.subsetsum_divide module¶

Subsetsum by splitting christoph dürr et jill-jênn vie - 2014-2019

tryalgo.subsetsum_divide.part_sum(x_table, i=0)[source]¶

All subsetsums from x_table[i:]

Parameters

x_table – table of values
i (int) – index_table defining suffix_table of x_table to be considered

Iterates

over all values, in arbitrary order

Complexity

$O(2^{len(x_table)-i})$

tryalgo.subsetsum_divide.part_sum2(x_table)[source]¶

All subsetsums from a list x

Parameters: x_table – list of values
Complexity: $O(2^{len(x)})$

tryalgo.subsetsum_divide.subset_sum(x_table, r_target)[source]¶

Subsetsum by splitting

Parameters

x_table – table of values
r_target – target value

Returns bool

if there is a subsequence of x_table with total sum r_target

Complexity

$O(n^{\lceil n/2 \rceil})$

tryalgo.subsetsum_divide.subset_sum2(x_table, r_target)[source]¶

Subsetsum by splitting

Parameters

x_table – table of values
r_target – target value

Returns bool

if there is a subsequence of x_table with total sum r_target

Complexity

$O(n^{\lceil n/2 \rceil})$

tryalgo.sudoku module¶

Solving Sudoku (nanpure) jill-jenn vie et christoph durr - 2014-2019

tryalgo.sudoku.sudoku(G)[source]¶

Solving Sudoku

Parameters: G – integer matrix with 0 at empty cells
Returns bool: True if grid could be solved
Modifies: G will contain the solution
Complexity: huge, but linear for usual published 9x9 grids

tryalgo.three_partition module¶

subsetsum jill-jênn vie et christoph dürr - 2015-2019

tryalgo.three_partition.three_partition(x)[source]¶

partition a set of integers in 3 parts of same total value

Parameters: x – table of non negative values
Returns: triplet of the integers encoding the sets, or None otherwise
Complexity: $O(2^{2n})$

tryalgo.topological_order module¶

Topological order jill-jênn vie et christoph dürr - 2014-2019

tryalgo.topological_order.topological_order(graph)[source]¶

Topological sorting by maintaining indegree

Parameters: graph – directed graph in listlist format, cannot be listdict
Returns: list of vertices in order
Complexity: O(|V|+|E|)

tryalgo.topological_order.topological_order_dfs(graph)[source]¶

Topological sorting by depth first search

Parameters: graph – directed graph in listlist format, cannot be listdict
Returns: list of vertices in order
Complexity: O(|V|+|E|)

tryalgo.trie module¶

trie - correcteur orthographique jill-jênn vie et christoph dürr - 2014-2019

tryalgo.trie.Trie(S)[source]¶

Parameters: S – set of words
Returns: trie containing all words from S
Complexity: linear in total word sizes from S

class tryalgo.trie.TrieNode[source]¶: Bases: object

tryalgo.trie.add(T, w, i=0)[source]¶

Parameters

T – trie
w (string) – word to be added to T

Returns

new trie consisting of w added into T

Complexity

O(len(w))

tryalgo.trie.search(T, dist, w, i=0)[source]¶: Searches for w[i:] in trie T with distance at most dist

tryalgo.trie.spell_check(T, w)[source]¶

Spellchecker

Parameters

T – trie encoding the dictionary
w – given word

Returns

a closest word from the dictionary

Complexity

linear if distance was constant

tryalgo.two_sat module¶

Solving 2-SAT boolean formulas jill-jenn vie et christoph durr - 2015-2019

tryalgo.two_sat.two_sat(formula)[source]¶

Solving a 2-SAT boolean formula

Parameters: formula – list of clauses, a clause is pair of literals over X1,…,Xn for some n. a literal is an integer, for example -1 = not X1, 3 = X3
Returns: table with boolean assignment satisfying the formula or None
Complexity: linear

tryalgo.union_rectangles module¶

Union of rectangles jill-jênn vie et christoph dürr - 2014-2019

class tryalgo.union_rectangles.CoverQuery(L)[source]¶

Bases: object

Segment tree to maintain a set of integer intervals and permitting to query the size of their union.

change(i, k, offset)[source]¶: when offset = +1, adds an interval [i, k], when offset = -1, removes it :complexity: O(log L)

cover()[source]¶

Returns: the size of the union of the stored intervals

tryalgo.union_rectangles.rectangles_contains_point(R, x, y)[source]¶: Decides if at least one of the given rectangles contains a given point either strictly or on its left or top border

tryalgo.union_rectangles.union_intervals(intervals)[source]¶

Size of the union of a set of intervals

Sweep from left to right. Maintain in a counter number of opened intervals minus number of closed intervals.

Parameters: intervals – Counter, which describes a multiset of intervals. an interval is a pair of values.
Returns: size of the union of those intervals
Complexity: $O(n \log n)$

tryalgo.union_rectangles.union_rectangles(R)[source]¶

Area of union of rectangles.

Sweep from top to bottom. Maintain in a set the horizontal projection of rectangles, for which the top border has been processed but not yet the bottom.

Parameters: R – list of rectangles defined by (x1, y1, x2, y2) where (x1, y1) is top left corner and (x2, y2) bottom right corner
Returns: area
Complexity: $O(n^2 \log n)$

tryalgo.union_rectangles.union_rectangles_fast(R)[source]¶

Area of union of rectangles

Parameters: R – list of rectangles defined by (x1, y1, x2, y2) where (x1, y1) is top left corner and (x2, y2) bottom right corner
Returns: area
Complexity: $O(n^2)$

tryalgo.union_rectangles.union_rectangles_fastest(R)[source]¶

Area of union of rectangles

Parameters: R – list of rectangles defined by (x1, y1, x2, y2) where (x1, y1) is top left corner and (x2, y2) bottom right corner
Returns: area
Complexity: $O(n \log n)$

tryalgo.union_rectangles.union_rectangles_naive(R)[source]¶

Area of union of rectangles

Parameters: R – list of rectangles defined by (x1, y1, x2, y2) where (x1, y1) is top left corner and (x2, y2) bottom right corner
Returns: area
Complexity: $O(n^3)$

tryalgo.windows_k_distinct module¶

All sliding windows containing k distinct elements jill-jenn vie et christoph durr - 2014-2018

tryalgo.windows_k_distinct.windows_k_distinct(x, k)[source]¶

Find all largest windows containing exactly k distinct elements

Parameters

x – list or string
k – positive integer

Yields

largest intervals [i, j) with len(set(x[i:j])) == k

Complexity

O(|x|)

Module contents¶

Tryalgo is a Python library of algorithms and data structures. It is implemented by Christoph Dürr (CNRS, Sorbonne Université, LIP6, Paris) and Jill-Jênn Vie (Inria, Lille) It is documented in the book “Programmation efficace, 128 Algorithmes qu’il faut avoir compris et codés en Python”, editor Ellipses.