We have a group of N items (represented by integers from 1 to N), and we know that there is some total order defined for these items. You may assume that no two elements will be equal (for all a, b: a<b or b<a). However, it is expensive to compare two items. Your task is to make a number of comparisons, and then output the sorted order. The cost of determining if a < b is given by the bth integer of element a of costs (space delimited), which is the same as the ath integer of element b. Naturally, you will be judged on the total cost of the comparisons you make before outputting the sorted order. If your order is incorrect, you will receive a 0. Otherwise, your score will be opt/cost, where opt is the best cost anyone has achieved and cost is the total cost of the comparisons you make (so your score for a test case will be between 0 and 1). Your score for the problem will simply be the sum of your scores for the individual test cases.
Floyd-Warshall算法描述
1)適用范圍：
a)APSP(All Pairs Shortest Paths)
b)稠密圖效果最佳
c)邊權(quán)可正可負
2)算法描述：
a)初始化：dis[u,v]=w[u,v]
b)For k:=1 to n
For i:=1 to n
For j:=1 to n
If dis[i,j]>dis[i,k]+dis[k,j] Then
Dis[I,j]:=dis[I,k]+dis[k,j]
c)算法結(jié)束：dis即為所有點對的最短路徑矩陣
3)算法小結(jié)：此算法簡單有效，由于三重循環(huán)結(jié)構(gòu)緊湊，對于稠密圖，效率要高于執(zhí)行|V|次Dijkstra算法。時間復雜度O(n^3)。
考慮下列變形：如(I,j)∈E則dis[I,j]初始為1,else初始為0，這樣的Floyd算法最后的最短路徑矩陣即成為一個判斷I,j是否有通路的矩陣。更簡單的，我們可以把dis設(shè)成boolean類型，則每次可以用“dis[I,j]:=dis[I,j]or(dis[I,k]and dis[k,j])”來代替算法描述中的藍色部分，可以更直觀地得到I,j的連通情況。
