[Introduction to Algorithms - Lecture Notes] 16.Greedy Algorithms and Minimum Spanning Tree

Graphs(review) CLRS, App B

 

Digraph(Directed graph) G = (V, E)

 ·Set V of vertices(singular vertex)

·Set E ⊆ V * V of edges

Undirected graph: E contains unordered pairs.

|E| = O(V^2). G connected => |E| > |V| - 1

log|E| = Θ(log V)

 

Graph representations

Adjacency matrix of G = (V, E), where V = {1, 2, 3, ..., n}, is the n * n matrix A given by A[i, j] = {1 if (i, j) ∈E 
　　　　　　　　　 {0 if (i, j) ∉ E

Θ(V^2) storage => dense representation

Adjacency list of V∈ A is the list Adj[V] of vertices adjacent to V.

Adj[1] = {2, 3}

Adj[2] = {3}

Adj[3] = {}

Adj[4] = {3}

|Adj[V]|

= {degree(v) (undirected graph)
   {outdegree (directed graph)

 

Handshaking Lemma (undirected graph)

Σdegree(v) = 2|E|        [v ∈ V]

for undirected grqaphs => adj-list representation uses Θ(E + V) storage same thing asymptotically for digraphs.

sparse representation: often better than adj matrix.b

 

Minimum spanning trees

Input: Connected, undirected graph G = (V, E) with weight function w: E -> R.

·for simplicity, assume all edges are distinct.

Output: A spanning tree T (connects all the vertices) of minimum weight.

w(T) = ∑w(u, v)   [(u, v) ∈ T]

Ex.

Optimal substructure

MST T:

(other edges not shown)

remove (u, v)∈ T, then T is partitioned into two subtrees T1 and T2.

theorem: T1 is MST for G1 = (V1, E1), the subgraph of G induced by vertices in T1.

V1 = vertices in T1

E1 = {(x, y) ∈ E: x, y ∈ V1}

Proof. Cut & Paste
w(T) = w(u, v) + w(T1) + w(T2)

If T1' better than T1 for G1, then T' = {(u, v) } U T1' U T2, would be better than T for F.

Overlapping subproblems? YES

Dynamic programing? Yes, but MST exhibits an even more powerful proberty.

Hallmark for greedy algorithms

Greedy choice proberty: A locally optimal choice is globally optimal.

 

Theorem: Let T be MST of G = (V, E), let A ⊆ V, suppose (u, v)∈E is the least-weight edge connecting A to V-A. Then (u, v)∈T.

Consider unique simple path from u to v in T. Swap (u, v) with the first edge on this path that connects a vertex V in A to a vertex in V - A. A lower weight spanning tree than T results. Contradiction.

 

Prim's algorithm

Idea: Maintain V-A as a priority queue Q, Key each vertex in Q with weight of least-weight edge connecting it to a vertex in A.

Q ← V
key[v] ← ∞ for all v ∈ V
key[s] ← 0 for some arbitrary s ∈ V
    while Q ≠ ∅
        do u ← EXTRACT-MIN(Q)
            for each v ∈ Adj[u]
                do if v ∈ Q and w(u, v) < key[v]
                    then key[v] ← w(u, v) ⊳ DECREASE-KEY
                        π[v] ← u  

At end, {(V, π[v])} forms MST.

Q                       TEXTRACT-MIN          TDECREASE-KEY         Total

array                  O(V)                          O(1)                           O(V2)

binary heap        O(lg V)                       O(lg V)                       O(E lg V)

Fibonacci heap    O(lg V)                       O(1)                           O(E + V lg V)
                         amortized                   amortized                   worst case