Codeforces 464E The Classic Problem (最短路 + 主席树 + hash）

题意及思路

这个题加深了我对主席树的理解，是个好题。每次更新某个点的距离时，是以之前对这个点的插入操作形成的线段树为基础，在O(logn）的时间中造出了一颗新的线段树，相比直接创建n颗线段树更省时间。比较的时候二分比较，为了加快比较给每个点设置一个hash值。

代码：

#include <bits/stdc++.h>
using namespace std;
const unsigned long long P = 13331;
const int mod = 1000000007;
const int maxn = 100010;

int head[maxn], Next[maxn * 2], edge[maxn * 2], ver[maxn * 2];
int tot, tote;
int b[maxn * 2], pre[maxn];
int mx, dis[maxn];
bool vis[maxn];
int Stack[maxn], Top;

void add(int x, int y, int z) {
	ver[++tote] = y;
	edge[tote] = z;
	Next[tote] = head[x];
	head[x] = tote;
}
struct SegementTree {
	int ls, rs;
	int sum;
	unsigned long long hash;
}tr[maxn * 200];

void pushup(int x) {
	tr[x].sum = tr[tr[x].ls].sum + tr[tr[x].rs].sum;
	unsigned long long tmp1 = tr[tr[x].ls].hash, tmp2 = tr[tr[x].rs].hash;
	tr[x].hash = (((tmp1 * P + tmp2) ^ tmp1) * P ^ tmp2) * P;
}

void insert(int &now, int l, int r, int pos) {
	int p = now;
	now = ++tot;
	tr[now] = tr[p];
	if(l == r) {
		tr[now].sum = 1;
		tr[now].hash = l + P;
		return;
	}
	int mid = (l + r) >> 1;
	if(pos <=  mid)  insert(tr[now].ls, l, mid, pos);
	else insert(tr[now].rs, mid + 1, r, pos);
	pushup(now);
}

void del(int &now, int l, int r, int ql, int qr) {
	if(l >= ql && r <= qr) {
		now = 0;
		return;
	}
	int p = now;now = ++tot;
	tr[now] = tr[p];
	int mid = (l + r) >> 1;
	if(ql <= mid) del(tr[now].ls, l, mid, ql, qr);
	if(qr > mid) del(tr[now].rs, mid + 1, r, ql, qr);
	pushup(now);
}

int query(int now, int l, int r, int pos) {
	if(l == r) {
		if(tr[now].sum == 0) return l;
		return -1;
	}
	int mid = (l + r) >> 1;
	if(pos > mid) return query(tr[now].rs, mid + 1, r, pos);
	int ans = query(tr[now].ls, l, mid, pos);
	if(ans != -1) return ans;
	return query(tr[now].rs, mid + 1, r, pos);
}

int add(int now, int pos) {
	int p = query(now, 0, mx, pos);
	if(p > pos) del(now, 0, mx, pos, p - 1);
	insert(now, 0, mx, p);
	return now;
}

int get_sum(int now, int l, int r) {
	int mid = (l + r) >> 1;
	if(now == 0) return 0;
	if(l == r) return b[l];
	return (get_sum(tr[now].ls, l, mid) + get_sum(tr[now].rs, mid + 1, r)) % mod;
}

bool cmp(int x, int y, int l, int r) {
	if(l == r) return tr[x].sum >= tr[y].sum;
	int mid = (l + r) >> 1;
	if(tr[tr[x].rs].hash != tr[tr[y].rs].hash) 
		return cmp(tr[x].rs, tr[y].rs, mid + 1, r);
	else return cmp(tr[x].ls, tr[y].ls, l, mid);
}

struct node {
	int x, y;
	bool operator < (const node& rhs) const {
		return cmp(x, rhs.x, 0, mx);
	}
};

priority_queue<node> q;

void dijkstra(int s) {
	dis[s] = ++tot, q.push((node){dis[s], s});
	while(!q.empty()) {
		node tmp = q.top();
		q.pop();
		if(vis[tmp.y]) continue;
		vis[tmp.y] = 1;
		int x = tmp.y;
		for (int i = head[x]; i; i = Next[i]) {
			int y = ver[i], z = add(dis[x], edge[i]);
			if(!dis[y] || !cmp(z, dis[y], 0, mx)) {
				pre[y] = x;
				dis[y] = z;
				q.push((node){z, y});
			}
		}
	}
}

void print(int x, int deep) {
	if(x == 0) {
		printf("%d\n", deep);
		return;
	} 
	print(pre[x], deep + 1);
	printf("%d ", x);
}
int main() {
	int n, m, x, y, z, s, t;
	scanf("%d%d", &n, &m);
	for (int i = 1; i <= m; i++) {
		scanf("%d%d%d", &x, &y, &z);
		add(x, y, z);
		add(y, x, z);
		mx = max(mx, z);
	}
	b[0] = 1;
	mx += 20;
	for (int i = 1; i <= mx; i++) {
		b[i] = (b[i - 1] << 1) % mod;
	}
	scanf("%d%d", &s, &t);
	dijkstra(s);
	if(dis[t] == 0) printf("-1\n");
	else {
		printf("%d\n", get_sum(dis[t], 0, mx));
		while(t) {
			Stack[++Top] = t;
			t = pre[t];
		}
		printf("%d\n", Top);
		while(Top) {
			printf("%d ", Stack[Top]);
			Top--;
		}
	}
}