Graph

Graph Representation

Graph Problems and Algorithms

In addition, we also have planar graphs, dynamic graphs, etc.

Traversal-based Algorithms Review

• Breadth-first Search (BFS)

• Depth-first Search (DFS)

• Topological Sort

• Graph Connectivity

  • for each vertex, if not visited, run BFS/DFS on it

• Algorithms based on DFS

  • Bi-connectivity, articulation points, bridges (CLRS pp. 621-622)

  • Topological sort (CLRS pp. 612-614)

  • Strongly connected components (CLRS pp. 615-618)

Minimum Spanning Tree (MST)

• undirected weighted graph

• find n-1 edges that connected all vertices with minimum total weights

Light-edge property

• The lightest edge in a cut must be in the MST

Prim's Algorithm

• start from an arbitrary vertex, add the lightest connected edge to the tree, and repeat

• this is a greedy algorithm

• implementation can be array-based, binary heap, or Fibonacci heap (CLRS Section 19)

Kruskal’s Algorithm

• scan edges by weight from lowest to highest, add the edge that doesn't introduce cycle, and repeat

• this is a greedy algorithm

• data structure: weighted edge list O(m log m) = O(m log n)

// linked forest + path compression 
find(x)     { return label[x] = x==label[x] ? x : find(label[x]); }
union(x, y) { if (find(x) != find(y)) label[label[y]] = label[x]; }
test(x, y)  { return (find(x) == find(y)); }

Single-Source Shortest Paths (SSSP)

• undirected graph, or directed acyclic graph (DAG)

• find a shortest path from a given source vertex u to each vertex v

Bellman-Ford Algorithm

• dynamic programming algorithm O(nm)

• can work for negative weight

• $D_{i,k} = \min \begin{cases} D_{i, k-1} \\ \min_{(j, i)\in E} \{D_{j, k-1} + w(j, i)\}\end{cases}$

// Bellman-Ford Algorithm
for i=1 to n do
  D[i]=MAX
D[s]=0
for k=1 to n-1 do
  for each (i,j) in E do
    if (D[i]+w(i,j)<D[j])
      D[j]=D[i]+w(i,j), from[j]=i
// optional optimization: if no distance is updated, break

Dijkstra's Algorithm

• start from an arbitrary node, expand the neighbor with the lowest cost, and repeat

• greedy algorithm (similar to Prim's)

• for positive weight only

• implementation: array-based O(n^2), binary heap O((m+n) log n), or Fibonacci heap (m + n log n)

All-Pairs Shortest Paths (APSP)

• find a shortest path from u to v for every pair of vertices u and v

• although we can solve this by running a single-source algorithm once from each vertex, we usually can solve it faster (e.g., by Floyd-Warshall algorithm)

Relax operation: update (u, v) path by weight w if better (CLRS pp. 648)

Strongly Connected Components (SCC)

• directed graph

• strongly connected: if every vertex is reachable from every other vertex

• find a partition such that each subgraph is strongly connected

Tarjan’s algorithm

• is a DFS-based linear-time algorithm O(n + m)

• kind of simple to implement, but counter-intuitive

• hard to parallel

  • DFS is P-Complete (DFS is inherently sequential [Reif 1985])

Reachability-based algorithm

• find SCC by reachability queries O(m log n)

  • what is the set of vertices that can be reached by a given vertex v

  • which vertices can reach v

  • the intersection of two sets form an SCC

• can be parallelized (BGSS 2016, Yan Gu, code)

• benchmark suite for graph algorithms

Bipartite Graph Matching

• bipartite graph: vertex set V can be partitioned into two subsets V1 and V2 such that each edge has one endpoint in V1 and the other endpoint in V2

• matching: a subset of the edges for which every vertex belongs to at most one of the edges

• maximum matching: a matching with the maximum number of edges

• perfect matching: a matching in which every vertex is matched

augmenting path

• a structure such that reversing path direction can get one more matched edge

  • unmatched vertex --- matched vertex --- ... --- matched vertex --- unmatched vertex

• how to find:

  • reverse the matched edges

  • start from an unmatched vertex, and try to find a path to another unmatched vertex

Hungarian algorithm

• find the maximum matching for bipartite graphs in O(nm) time

• find an augmenting path, reverse direction, get one more matched edge, and repeat

// a DFS-based implementation
// g[i][j] is the graph, p[j]=i stores the matched edge, 
// and b[j] is a temp boolean array
bool find(i)
  for j=1 to m do
    if g[i][j] == 1 and b[j] == false
      b[j] = true
      if p[j] == -1 or find(p[j])
        p[j]=i, return true
return false
Hungarian()
  p[1..n] = -1
  for i=1 to n do
    b[1..n] = false, find(i)

Maximum Flow (Flow Network)

• directed graph, with non-negative weight/capacity

• find the max-flow from the source s to the sink t

Ford-Fulkerson method

• greedy algorithm

• the notion of augmenting path

  • here we overload the notion of augmenting path on a general directed graph (instead of bipartite)

  • it is called augmenting path because once found, we can get additional flow from source to sink

• method: keep finding augmenting paths in the residual graph

  • start with finding an arbitrary path from s to t

  • reverse it, resulting in a residual graph

  • find a new path from s to t on the residual graph, and reverse it to generate a new residual graph

  • repeat until no paths can be found, then the sum of capacities of all found paths is the max-flow

• It is called "method" but not "algorithm" because how to find augmenting path is missing

  • depending on the actual method to find augmenting path, the algorithm efficiency can differ

Ways to find augmenting paths in a general directed graph

Dinic's Algorithm

• new concepts: level graph

  • construct level graph: BFS from source to every node and compute hops

  • then find augmenting paths on level graphs

• proposed by Dinic [1970]; fastest in practice O(n^2 m)

• on bipartite graph, this algorithm is referred to as Hopcroft-Karp algorithm [1973]

Related Discussions on Flow Network

Min-cost Flow Problem

• in a flow network, each edge also has a cost

• the goal is to maximizing the flow while minimizing the cost

Relationship to Linear Programming

• most combinatorial algorithms can be represented in a LP form (and solved by LP solver efficiently)

• we can convert a network flow problem into a LP problem

  • by formulating edges into linear constraints

• we can also convert a LP problem into a network flow problem

  • by adding a virtual node, for example

  • the flow property can promise the equality constraints

The dual of max matching: min vertex cover

• vertex cover: a set of vertices that includes at least one endpoint of every edge

• in any bipartite graph, maximum matching = minimum vertex cover [Konig and Egercary 1931]

The dual of max flow: min cut

• cut: a subset of edges that disconnects the source and the sink

• minimum cut: the cut with minimum sum of edge capacities

• in a flow network, maximum flow = minimum cut [Menger 1927]

Summary

• MST algorithms: Prim’s, Kruskal’s

• SSSP algorithms: Dijkstra’s, Bellman-Ford

• SCC algorithms: Tarjan’s algorithm, BGSS algorithm

• Bipartite graph matching: Hungarian algorithm

• Max flow: Ford-Fulkerson method

Example Code: Graph and BFS

#include <iostream>
#include <vector>
#include <array>
#include <list>
#include <queue>
#include <stack>
#include <set>
#include <map>
#include <unordered_set>
#include <unordered_map>
#include <cmath>
#include <functional>


class MyGraph { 
  private:
    // indices of vertices may not be consecutive
    std::set<int> vertices;                  // std::set, std::map -> O(log(n)) using red-black tree
    std::map<int, std::list<int>> edgeLists; // std::unordered_set, unordered_map -> O(1) using hashing
                                             // switching to unordered_set/map can speed up 600ms --> 400ms
  public:
    MyGraph(){};
    ~MyGraph(){};
    void addEdge(int src, int dest);
    void printGraph();
    void BFS(int startVertex, int& endVertex, int& depth);
};

void MyGraph::addEdge(int src, int dest) {
  vertices.insert(src);
  vertices.insert(dest);
  edgeLists[src].push_back(dest);
  edgeLists[dest].push_back(src);  // undirected graph 
}

void MyGraph::printGraph(){
  std::cout << "vertices:" << std::endl;
  for (const auto& v : vertices)
    std::cout << v << " ";
  std::cout << std::endl;
  std::cout << "edges in adjacency lists:" << std::endl;
  for (const auto& v : vertices){
    std::cout << v << ": ";
    for (const auto& e : edgeLists[v])
      std::cout << e << " "; 
    std::cout << std::endl;
  }
  std::cout << std::endl;
}

void MyGraph::BFS(int startVertex, int& endVertex, int& depth) {
  std::unordered_map<int, bool> visited (vertices.size());  // default to false
  std::unordered_map<int, int> distance (vertices.size());
  std::vector<int> visited_list;
  std::queue<int> queue;
  visited[startVertex] = true;
  distance[startVertex] = 0;
  queue.push(startVertex);
  while (!queue.empty()) {
    int currVertex = queue.front();
    queue.pop();
    visited_list.push_back(currVertex);
    for (const auto& adjVertex : edgeLists[currVertex])
      if (!visited[adjVertex]) {
        visited[adjVertex] = true;
        distance[adjVertex] = distance[currVertex] + 1;
        queue.push(adjVertex);
      }
  }
  endVertex = visited_list.back();
  depth = distance[endVertex];
}

int main(){
  int n;
  std::cin >> n; 
  MyGraph graph;
  int i, j;
  for (int k = 0; k < n-1; ++k){
    std::cin >> i >> j;
    graph.addEdge(i, j);
  }

  // run BFS twice to find the diameter of the graph
  // can be proved by triangle inequality
  int startVertex, endVertex, depth;
  startVertex = i;
  graph.BFS(startVertex, endVertex, depth);

  startVertex = endVertex;
  graph.BFS(startVertex, endVertex, depth);

  std::cout << (depth+1)/2 << std::endl;  // ceiling

  return 0;
}