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Shortest paths and spanning trees in graphs
Lyzhin Ivan,
2015
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Shortest path problem
The problem of finding a path
between two vertices such that the sum of the
weights of edges in path is minimized.
Known algorithms:
Dijkstra
Floyd–Warshall
Bellman–Ford
and so on...
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Dijkstra algorithm
There are two sets of vertices –
visited and unvisited.
For visited vertices we know minimal distance
from start. For unvisited vertices we know some distance which can be not minimal.
Initially, all vertices are unvisited and distance to each vertex is INF. Only distance to start node is equal 0.
On each step choose unvisited vertex with minimal distance. Now it’s visited vertex. And try to relax distance of neighbors.
Complexity: trivial implementation O(|V|^2+|E|)
implementation with set O(|E|log|V|+|V|log|V|)
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Trivial implementation
void dijkstra(int s)
{
vector mark(n, false);
vector d(n, INF);
d[s]
= 0;
for (int i = 0; i < n;
++i)
{
int u = -1;
for (int j = 0; j < n; ++j)
if (!mark[j] && (u == -1 || d[j] < d[u]))
u = j;
mark[u] = true;
for (v - сосед u)
d[v] = min(d[v], d[u] + weight(uv));
}
}
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Implementation with set
void dijkstra(int s)
{
set q;
//(dist[u], u)
vector dist(n, INF);
dist[s] = 0;
q.insert(mp(0, s));
while(!q.empty())
{
int cur =
q.begin()->second;
q.erase(q.begin());
for(auto e : g[cur])
if(dist[e.first] > dist[cur]+e.second)
{
q.erase(mp(dist[e.first], e.first));
dist[e.first] = dist[cur] + e.second;
q.insert(mp(dist[e.first], e.first));
}
}
}
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Implementation with priority queue
void dijkstra(int s)
{
priority_queue
q; //(dist[u], u)
vector dist(n, INF);
dist[s] = 0;
q.push(mp(0, s));
while(!q.empty())
{
int cur
= q.top().second;
int cur_d = -q.top().first; q.pop();
if(cur_d > dist[cur]) continue;
for(auto e : g[cur])
if(dist[e.first] > dist[cur]+e.second)
{
dist[e.first] = dist[cur] + e.second;
q.push(mp(-dist[e.first], e.first));
}
}
}
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Floyd–Warshall algorithm
Initially, dist[u][u]=0 and for each edge (u,
v): dist[u][v]=weight(u, v)
On iteration k we let use vertex
k as intermediate vertex and for each pair of vertices we try to relax distance.
dist[u][v] = min(dist[u][v], dist[u][k]+dist[k][v])
Complexity: O(|V|^3)
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Implementation
void floyd_warshall()
{
vector dist(n, vector(n, INF));
for (int i
= 0; i < n; ++i)
dist[i][i] = 0;
for (int
i = 0; i < n; ++i)
for (auto e : g[i])
dist[i][e.first] = e.second;
for (int k = 0; k < n; ++k)
for (int i = 0; i < n; ++i)
for (int j = 0; j < n; ++j)
dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j]);
}
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Bellman–Ford algorithm
|V|-1 iterations, on each we try relax
distance with all edges.
If we can relax distance on
|V| iteration then negative cycle exists in graph
Why |V|-1 iterations? Because the longest way without cycles from one node to another one contains no more |V|-1 edges.
Complexity O(|V||E|)
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Implementation
void bellman_ford(int s)
{
vector dist(n, INF);
dist[s] = 0;
for (int
i = 0; i < n - 1; ++i)
for
(auto e : edges)
dist[e.v] = min(dist[e.v], dist[e.u] + e.weight);
for (auto e : edges)
if (dist[e.v] > dist[e.u] + e.weight)
cout << "Negative cycle!" << endl;
}
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Minimal spanning tree
A spanning tree T of an
undirected graph G is a subgraph that includes all
of the vertices of G that is a tree.
A minimal spanning tree is a spanning tree and sum of weights is minimized.
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Prim’s algorithm
Initialize a tree with a single vertex,
chosen arbitrarily from the graph.
Grow the tree by one
edge: of the edges that connect the tree to vertices not yet in the tree, find the minimum-weight edge, transfer it to the tree and try to relax distance for neighbors.
Repeat step 2 (until all vertices are in the tree).
Complexity: trivial implementation O(|V|^2+|E|)
implementation with set O(|E|log|V|+|E|)
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Implementation
void prima()
{
set q; //(dist[u], u)
vector dist(n,
INF);
dist[0] = 0;
q.insert(mp(0, 0));
while (!q.empty())
{
int cur = q.begin()->second;
q.erase(q.begin());
for (auto
e : g[cur])
if (dist[e.first] > e.second)
{
q.erase(mp(dist[e.first], e.first));
dist[e.first] = e.second;
q.insert(mp(dist[e.first], e.first));
}
}
}
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Kruskal’s algorithm
Create a forest F (a set of
trees), where each vertex in the graph is a
separate tree
Create a set S containing all the edges in the graph
While S is nonempty and F is not yet spanning:
remove an edge with minimum weight from S
if the removed edge connects two different trees then add it to the forest F, combining two trees into a single tree
Complexity: trivial O(|V|^2+|E|log|E|)
with DSU O(|E|log|E|)
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Trivial implementation
void trivial_kruskal()
{
vector color(n);
for (int i = 0;
i < n; ++i)
color[i] = i;
sort(edges.begin(), edges.end());
for(auto e :
edges)
if(color[e.u]!=color[e.v])
{
add_to_spanning_tree(e);
int c1 = color[e.u];
int c2 = color[e.v];
for (int i = 0; i < n; ++i)
if (color[i] == c1)
color[i] = c2;
}
}
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Implementation with DSU
void kruskal()
{
DSU dsu(n);
sort(edges.begin(), edges.end());
for(auto e :
edges)
if(dsu.findSet(e.u)!=dsu.findSet(e.v))
{
add_to_spanning_tree(e);
dsu.unionSets(e.u, e.v);
}
}
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Disjoint-set-union (DSU)
Two main operations:
Find(U) – return root of
set, which contains U, complexity O(1)
Union(U, V) – join
sets, which contain U and V, complexity O(1)
After creating DSU:
After some operations:
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Implementation
struct DSU
{
vector p;
DSU(int n) {
p.resize(n);
for (int i =
0; i < n; ++i)
p[i] = i;
}
int find(int u)
{
return u == p[u] ? u : find(p[u]);
}
void merge(int u, int v) {
int pu = find(u);
int pv = find(v);
p[pv] = pu;
}
};
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Path compression
When we go up, we can remember
root of set for each vertex in path
int findSet(int
u)
{
return u == p[u] ? u : p[u] = findSet(p[u]);
}
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Union by size
int unionSets(int u, int v)
{
int pu
= findSet(u);
int pv = findSet(v);
if (pu == pv) return;
if
(sizes[pu] < sizes[pv])
swap(pu, pv);
p[pv] = pu;
sizes[pu] += sizes[pv];
}
DSU(int size)
{
p.resize(size);
sizes.resize(size, 1);
for (int i = 0; i < size; ++i)
p[i] = i;
}
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Links
https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm
https://en.wikipedia.org/wiki/Floyd–Warshall_algorithm
https://en.wikipedia.org/wiki/Bellman–Ford_algorithm
https://en.wikipedia.org/wiki/Kruskal%27s_algorithm
https://en.wikipedia.org/wiki/Prim%27s_algorithm
https://en.wikipedia.org/wiki/Disjoint-set_data_structure
http://e-maxx.ru/algo/topological_sort