Dijkstra 未優化版, 算法相對清晰:
// 關鍵1: 處理每個人的地位等級
// 辦法: 枚舉--假設某種方案是最省錢的,
// 則該方案中的所有交易者的地位等級都會落在一個寬度為rankLimit的區間
// 于是可以枚舉這個區間:
// [ownerRank[1] - rankLimit, ownerRank] ~ [ownerRank[1], ownerRank + rankLimit]
// 于是這道題考察了最短路的dijkstra算法與枚舉的結合.
//
// 其中枚舉可行是需要考察其復雜度的:
// dijkstra算法的復雜度為: O(n * n), n為節點數目
// 枚舉量為 rankLimit + 1;
// 于是枚舉 + dijkstra的算法復雜度為 O(n * n) * (rankLimit + 1)
// 關鍵2: 由題意要聯想到用最短路, 而且是邊權為正的最短路
// 1) 以物品為圖節點
// 2) 設i物品如果能用j物品以價格m交換, 則邊(i,j)的權值為m
// 3) 設求得節點1到物品x的最短路, 該最短路的權值和為tw(total weight的縮寫),
// 則從物品x開始物物交換的所有方案中, 最節省的方案會耗費tw + price[x]的金錢
// 而婚禮最少需要的金幣數就是所有 1 <= x <= goodsCount 中,
// tw[1][x] + price[x]最小的那個. (tw[1][x]表示1到x的最短路徑權值)
// 優化1: 在dijkstra算法的代碼部分, 需要對原點到節點的最小距離是否已知作出判斷.
// 這個判斷是用bool數組disKnown來判斷的, 浪費大量時間.
// 可以優化為添加一個數組, 用該數組保存最小距離未知的節點的編號.
// 只處理數組中的節點.
#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;
// 如果某個節點/商品擁有者的階級地位不在[rankStart, rankStart + rankLimit]
// 的范圍內, 就不必考慮該節點
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;
}
優化后, 速度要快一些, 但是代碼比較難看, 對變量的命名讓人比較惱火:
#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) 實質就是是確定圖中是否存在正環--沿著這個環一直走能夠使環內節點的權值無限增長.2) 特點是: 每個邊的權值是動態變化的, 這點提醒我們適合用Bellman Ford算法來處理(SPFA實質上是Bellman Ford的優化, 算法思想是一樣的).SPFA算法:
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