Dijkstra 未优化版, 算法相对清晰:
// 关键1: 处理每个人的地位等级
// 办法: 枚举--假设某种方案是最省钱的,
// 则该方案中的所有交易者的地位等级都会落在一个宽度为rankLimit的区间
// 于是可以枚举这个区间:
// [ownerRank[1] - rankLimit, ownerRank] ~ [ownerRank[1], ownerRank + rankLimit]
// 于是这道题考察了最短路的dijkstra算法与枚举的结合.
//
// 其中枚举可行是需要考察其复杂度的:
// dijkstra算法的复杂度为: O(n * n), n为节点数目
// 枚举量为 rankLimit + 1;
// 于是枚举 + dijkstra的算法复杂度为 O(n * n) * (rankLimit + 1)
// 关键2: 由题意要联想到用最短路, 而且是边权为正的最短路
// 1) 以物品为图节点
// 2) 设i物品如果能用j物品以价格m交换, 则边(i,j)的权值为m
// 3) 设求得节点1到物品x的最短路, 该最短路的权值和为tw(total weight的缩写),
// 则从物品x开始物物交换的所有方案中, 最节省的方案会耗费tw + price[x]的金钱
// 而婚礼最少需要的金币数就是所有 1 <= x <= goodsCount 中,
// tw[1][x] + price[x]最小的那个. (tw[1][x]表示1到x的最短路径权值)
// 优化1: 在dijkstra算法的代码部分, 需要对原点到节点的最小距离是否已知作出判断.
// 这个判断是用bool数组disKnown来判断的, 浪费大量时间.
// 可以优化为添加一个数组, 用该数组保存最小距离未知的节点的编号.
// 只处理数组中的节点.
#include <cstdio>
using namespace std;
struct Node {
int to;
int weight;
Node *next;
};
#define INF (1 << 30)
#define MAXNODE (100 + 10)
#define MAXEDGE (MAXNODE * MAXNODE + 10)
Node nodeHead[MAXNODE + 1];
Node nodes[MAXEDGE];
int ownerRank[MAXNODE + 1];
int price[MAXNODE + 1];
int minWeight[MAXNODE + 1];
bool disKnown[MAXNODE + 1];
int allocPos = 0;
Node *getNode() {
return nodes + allocPos++;
}
void initGraph(int n) {
allocPos = 0;
int i = 0;
for (i = 0; i < n; ++i) {
nodeHead[i].next = NULL;
minWeight[i] = INF;
}
}
void addEdge(int from, int to, int weight) {
Node *newNode = getNode();
newNode->next = nodeHead[from].next;
newNode->to = to;
newNode->weight = weight;
nodeHead[from].next = newNode;
}
int main() {
int rankLimit, goodsCount, substituteCount, subPrice, num, minPrice, minWei;
int minWeiPos;
int i, j, rankStart;
scanf("%d%d", &rankLimit, &goodsCount);
initGraph(goodsCount + 1);
for (i = 1; i <= goodsCount; ++i) {
scanf("%d%d%d", price + i, ownerRank + i, &substituteCount);
for (j = 0; j < substituteCount; ++j) {
scanf("%d%d", &num, &subPrice);
addEdge(i, num, subPrice);
}
}
minPrice = price[1];
for (rankStart = ownerRank[1] - rankLimit; rankStart <= ownerRank[1]; rankStart++) {
for (i = 1; i <= goodsCount; ++i) {
minWeight[i] = INF;
// 如果某个节点/商品拥有者的阶级地位不在[rankStart, rankStart + rankLimit]
// 的范围内, 就不必考虑该节点
if (ownerRank[i] < rankStart || ownerRank[i] > rankStart + rankLimit) {
disKnown[i] = true;
}
else {
disKnown[i] = false;
}
}
disKnown[1] = false;
minWeight[1] = 0;
for (i = 1; i <= goodsCount; ++i) {
minWei = INF;
for (j = 1; j <= goodsCount; ++j) {
if (!disKnown[j] && minWeight[j] < minWei) {
minWei = minWeight[j];
minWeiPos = j;
}
}
disKnown[minWeiPos] = true;
if (minWei + price[minWeiPos] < minPrice) {
minPrice = minWei + price[minWeiPos];
}
for (Node *tra = nodeHead[minWeiPos].next; tra != NULL; tra = tra->next) {
if (!disKnown[tra->to] &&
minWeight[tra->to] > minWeight[minWeiPos] + tra->weight ) {
minWeight[tra->to] = minWeight[minWeiPos] + tra->weight;
}
}
}
}
printf("%d\n", minPrice);
return 0;
}
优化后, 速度要快一些, 但是代码比较难看, 对变量的命名让人比较恼火:
#include <cstdio>
using namespace std;
struct Node {
int to;
int weight;
Node *next;
};
#define INF (1 << 30)
#define MAXNODE (100 + 10)
#define MAXEDGE (MAXNODE * MAXNODE + 10)
Node nodeHead[MAXNODE + 1];
Node nodes[MAXEDGE];
int ownerRank[MAXNODE + 1];
int price[MAXNODE + 1];
int minWeight[MAXNODE + 1];
int distanceUnknown[MAXNODE + 1];
int distanceUnknownCount;
bool isDistanceKnown[MAXNODE + 1];
int allocPos = 0;
Node *getNode() {
return nodes + allocPos++;
}
void initGraph(int n) {
allocPos = 0;
int i = 0;
for (i = 0; i < n; ++i) {
nodeHead[i].next = NULL;
minWeight[i] = INF;
}
}
void addEdge(int from, int to, int weight) {
Node *newNode = getNode();
newNode->next = nodeHead[from].next;
newNode->to = to;
newNode->weight = weight;
nodeHead[from].next = newNode;
}
int main() {
int rankLimit, goodsCount, substituteCount, subPrice, num, minPrice, minWei;
int minWeiDisUnkPos;
int i, j, from;
scanf("%d%d", &rankLimit, &goodsCount);
initGraph(goodsCount + 1);
for (i = 1; i <= goodsCount; ++i) {
scanf("%d%d%d", price + i, ownerRank + i, &substituteCount);
for (j = 0; j < substituteCount; ++j) {
scanf("%d%d", &num, &subPrice);
addEdge(i, num, subPrice);
}
}
minPrice = price[1];
for (from = ownerRank[1] - rankLimit; from <= ownerRank[1]; from++) {
for (i = 1; i <= goodsCount; ++i) {
minWeight[i] = INF;
}
distanceUnknownCount = 0;
for (i = 1; i <= goodsCount; ++i) {
if (ownerRank[i] >= from && ownerRank[i] <= from + rankLimit) {
distanceUnknown[distanceUnknownCount++] = i;
isDistanceKnown[i] = false;
}
else {
isDistanceKnown[i] = true;
}
}
minWeight[1] = 0;
isDistanceKnown[1] = false;
int n = distanceUnknownCount;
for (i = 0; i < n; ++i) {
minWei = INF;
for (j = 0; j < distanceUnknownCount; ++j) {
if (minWeight[ distanceUnknown[j] ] < minWei) {
minWei = minWeight[ distanceUnknown[j] ];
minWeiDisUnkPos = j;
}
}
if (minWei + price[ distanceUnknown[minWeiDisUnkPos] ] < minPrice) {
minPrice = minWei + price[ distanceUnknown[minWeiDisUnkPos] ];
}
for (Node *tra = nodeHead[ distanceUnknown[minWeiDisUnkPos] ].next; tra != NULL; tra = tra->next) {
if (!isDistanceKnown[tra->to] &&
minWeight[tra->to] > minWeight[ distanceUnknown[minWeiDisUnkPos] ] + tra->weight ) {
minWeight[tra->to] = minWeight[ distanceUnknown[minWeiDisUnkPos] ] + tra->weight;
}
}
isDistanceKnown[ distanceUnknown[minWeiDisUnkPos] ] = true;
distanceUnknown[minWeiDisUnkPos] = distanceUnknown[--distanceUnknownCount];
}
}
printf("%d\n", minPrice);
return 0;
}
摘要: 1) 实质就是是确定图中是否存在正环--沿着这个环一直走能够使环内节点的权值无限增长.2) 特点是: 每个边的权值是动态变化的, 这点提醒我们适合用Bellman Ford算法来处理(SPFA实质上是Bellman Ford的优化, 算法思想是一样的).SPFA算法:
Code highlighting produced by Actipro CodeHighlighter (freeware...
阅读全文