版本 4af44e2397612a75ef2ef9adc9430d5c19b19ae3

acm/course/Shortest_Path

Changes from 4af44e2397612a75ef2ef9adc9430d5c19b19ae3 to 55b5f2b4ad2cd481698150310afa65231d824914

Shortest Path
========
Single Source Shortest Path
--------
- <big>若使用Backtracking演算法，將枚舉所有路徑，必會TLE，因此我們將列舉幾項好用DER演算法!!</big>

![](/acm/13125075_944400135668382_1681010432_n.gif)


###Relaxation
- <big>若點v離source的距離為8、點u離source的距離為3，現在有一個點k，(u,k)長度為2、(k,v)長度為1，則對(u,v)做Relax。Relax後點v離source距離更改為6。</big>

![](/acm/Relax.png)

~~~{.c }
Relex(u,v,w){
   if(dis[u]+w(u,v)<dis[v])
       dis[v]=dis[u]+w(u,v);
}
~~~


###Bellman Ford
- <big>對所有的邊做n-1次的Relax</big>
- 時間複雜度：O(VE)

~~~{.c }
//Pseudo Code
BellmanFord(){
    //init
    dis[source]=0;
    for(all i != source) 
        dis[i] = INF;
    //n-1 times
    for(i=0;i<n-1;i++)
        for(each edge w(u,v) in G)
            Relax(u,v,w);
}
~~~

- <big>若有負的cycle，每做一次Relax，source至cycle上的點，就會持續變小。</big>

###SPFA()

All Pair Shortest Path
---------
###Floyd