图类算法总结

对最近学习的基础算法做一些总结，并举出一些相应的例子，仅此作为记录。
(未完待续…)

图的遍历（Graph Traversal)

DFS:

function DFS(hV, Ei)
	mark each node in V with 0
	count ← 0
	for each v in V do
		if v is marked 0 then
		DfsExplore(v )
function DfsExplore(v )
	count ← count + 1
	mark v with count
	for each edge (v , w) do ⊲ w is v ’s neighbour
		if w is marked with 0 then
		DfsExplore(w)

BFS:

function BFS(hV, Ei)
	mark each node in V with 0
	count ← 0, init(queue) ⊲ create an empty queue
	for each v in V do
		if v is marked 0 then
			count ← count + 1
			mark v with count
			inject(queue, v ) ⊲ queue containing just v
			while queue is non-empty do
				u ← eject(queue) ⊲ dequeues u
				for each edge (u, w) adjacent to u do
					if w is marked with 0 then
						count ← count + 1
						mark w with count
						inject(queue, w) ⊲ enqueues w

Topological Sorting：

最短路径

DP
核心思想：选取中间点
Warshall

for k ← 1 to n do
	for i ← 1 to n do
		if A[i, k] then
			for j ← 1 to n do
				if A[k, j] then
					A[i, j] ← 1

Floyd
基本思想：
(1) 最开始只允许经过1号顶点进行中转；

(2) 接下来只允许经过1和2号顶点进行中转，以此类推；

(3) 直到最后允许经过1~n号所有顶点进行中转，求任意两点之间的最短路程。

function Floyd(W [1..n, 1..n])
	D ← W
	for k ← 1 to n do
		for i ← 1 to n do
			for j ← 1 to n do
				D[i, j] ← min(D[i, j], D[i, k] + D[k, j])
	return D

greedy
核心思想：选取目前最短的路径
Prim
最小生成树(minimum spanning tree)

function Prim(hV, Ei)
	for each v ∈ V do
		cost[v ] ← ∞
		prev [v ] ← nil
	pick initial node v0
	cost[v0] ← 0
	Q ← InitPriorityQueue(V) //全结点入队
	while Q is non-empty do
		u ← EjectMin(Q)//逐个出队
		for each (u, w) ∈ E do
			if weight(u, w) < cost[w] then
				cost[w] ← weight(u, w)
				prev [w] ← u
				Update(Q, w, cost[w])

Dijkstra

function Dijkstra(hV, Ei, v0)
	for each v ∈ V do
		dist[v ] ← ∞
		prev [v ] ← nil
	dist[v0] ← 0
	Q ← InitPriorityQueue(V)//全结点入队
	while Q is non-empty do
		u ← EjectMin(Q)//逐个出队
		for each (u, w) ∈ E do
			if dist[u] + weight(u, w) < dist[w] then
				dist[w] ← dist[u] + weight(u, w)
				prev [w] ← u
				Update(Q, w, dist[w])