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Commit 0c09215

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added shortest path; added prim's algorithm to heap; improved variables' names
1 parent db763bd commit 0c09215

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4 files changed

+159
-42
lines changed

4 files changed

+159
-42
lines changed

‎basic_algorithm/graph/README.md

Lines changed: 3 additions & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -4,4 +4,6 @@ Ongoing...
44

55
[拓扑排序](./topological_sorting.md)
66

7-
[最小生成树](./mst.md)
7+
[最小生成树](./mst.md)
8+
9+
[最短路径](./shortest_path.md)

‎basic_algorithm/graph/mst.md

Lines changed: 39 additions & 39 deletions
Original file line numberDiff line numberDiff line change
@@ -9,10 +9,10 @@
99
```Python
1010
# Kruskal's algorithm
1111
class Solution:
12-
def getMinRiskValue(self, n, m, x, y, w):
12+
def getMinRiskValue(self, N, M, X, Y, W):
1313

1414
# Kruskal's algorithm with union-find to construct MST
15-
parent = list(range(n + 1))
15+
parent = list(range(N + 1))
1616

1717
def find(x):
1818
if parent[parent[x]] != parent[x]:
@@ -27,68 +27,68 @@ class Solution:
2727
else:
2828
return False
2929

30-
edges = sorted(zip(x, y, w), key=lambdax: x[2])
30+
edges = sorted(zip(W, X, Y))
3131

32-
mst_edges = []
32+
MST_edges = []
3333
for edge in edges:
34-
if union(edge[0], edge[1]):
35-
mst_edges.append(edge)
36-
if find(0) == find(n):
34+
if union(edge[1], edge[2]):
35+
MST_edges.append(edge)
36+
if find(0) == find(N):
3737
break
3838

39-
mst = collections.defaultdict(list)
39+
MST = collections.defaultdict(list)
4040
target = find(0)
41-
for u, v, r in mst_edges:
41+
for w, u, vin MST_edges:
4242
if find(u) == target and find(v) == target:
43-
mst[u].append((v, r))
44-
mst[v].append((u, r))
43+
MST[u].append((v, w))
44+
MST[v].append((u, w))
4545

4646
# dfs to search route from 0 to n
4747
dfs = [(0, None, float('-inf'))]
4848
while dfs:
49-
v, p, max_risk = dfs.pop()
50-
for a, r in mst[v]:
51-
cur_max = max(max_risk, r)
52-
if a == n:
53-
return cur_max
54-
if a != p:
55-
dfs.append((a, v, cur_max))
49+
v, p, max_w = dfs.pop()
50+
for n, w in MST[v]:
51+
cur_max_w = max(max_w, w)
52+
if n == N:
53+
return cur_max_w
54+
if n != p:
55+
dfs.append((n, v, cur_max_w))
5656
```
5757

5858
```Python
5959
# Prim's algorithm
6060
class Solution:
61-
def getMinRiskValue(self, n, m, x, y, w):
61+
def getMinRiskValue(self, N, M, X, Y, W):
6262

6363
# construct graph
6464
adj = collections.defaultdict(list)
65-
for i in range(m):
66-
adj[x[i]].append((y[i], w[i]))
67-
adj[y[i]].append((x[i], w[i]))
65+
for i in range(M):
66+
adj[X[i]].append((Y[i], W[i]))
67+
adj[Y[i]].append((X[i], W[i]))
6868

6969
# Prim's algorithm with min heap
70-
mst = collections.defaultdict(list)
71-
min_heap = [(r, 0, v) for v, r in adj[0]]
70+
MST = collections.defaultdict(list)
71+
min_heap = [(w, 0, v) for v, w in adj[0]]
7272
heapq.heapify(min_heap)
7373

74-
while n not in mst:
75-
r, u, v = heapq.heappop(min_heap)
76-
if v not in mst:
77-
mst[u].append((v, r))
78-
mst[v].append((u, r))
79-
for nei, w in adj[v]:
80-
if nei not in mst:
81-
heapq.heappush(min_heap, (w, v, nei))
74+
while N not in MST:
75+
w, p, v = heapq.heappop(min_heap)
76+
if v not in MST:
77+
MST[p].append((v, w))
78+
MST[v].append((p, w))
79+
for n, w in adj[v]:
80+
if n not in MST:
81+
heapq.heappush(min_heap, (w, v, n))
8282

8383
# dfs to search route from 0 to n
8484
dfs = [(0, None, float('-inf'))]
8585
while dfs:
86-
v, p, max_risk = dfs.pop()
87-
for a, r in mst[v]:
88-
cur_max = max(max_risk, r)
89-
if a == n:
90-
return cur_max
91-
if a != p:
92-
dfs.append((a, v, cur_max))
86+
v, p, max_w = dfs.pop()
87+
for n, w in MST[v]:
88+
cur_max_w = max(max_w, w)
89+
if n == N:
90+
return cur_max_w
91+
if n != p:
92+
dfs.append((n, v, cur_max_w))
9393
```
9494

‎basic_algorithm/graph/shortest_path.md

Lines changed: 69 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,69 @@
1+
# 最短路径问题
2+
3+
## Dijkstra's Algorithm
4+
5+
思想是 greedy 构造 shortest path tree (SPT),每次将当前距离源点最短的不在 SPT 中的结点加入SPT,与构造最小生成树 (MST) 的 Prim's algorithm 非常相似。可以用 priority queue (heap) 实现。
6+
7+
### [network-delay-time](https://leetcode-cn.com/problems/network-delay-time/)
8+
9+
标准的单源最短路径问题,使用朴素的的 Dijikstra 算法即可,可以当成模板使用。
10+
11+
```Python
12+
class Solution:
13+
def networkDelayTime(self, times: List[List[int]], N: int, K: int) -> int:
14+
15+
# construct graph
16+
graph_neighbor = collections.defaultdict(list)
17+
for s, e, t in times:
18+
graph_neighbor[s].append((e, t))
19+
20+
# Dijkstra
21+
SPT = {}
22+
min_heap = [(0, K)]
23+
24+
while min_heap:
25+
delay, node = heapq.heappop(min_heap)
26+
if node not in SPT:
27+
SPT[node] = delay
28+
for n, d in graph_neighbor[node]:
29+
if n not in SPT:
30+
heapq.heappush(min_heap, (d + delay, n))
31+
32+
return max(SPT.values()) if len(SPT) == N else -1
33+
```
34+
35+
### [cheapest-flights-within-k-stops](https://leetcode-cn.com/problems/cheapest-flights-within-k-stops/)
36+
37+
在标准的单源最短路径问题上限制了路径的边数,因此需要同时维护当前 SPT 内每个结点最短路径的边数,当遇到边数更小的路径 (边权和可以更大) 时结点需要重新入堆,以更新后继在边数上限内没达到的结点。
38+
39+
```Python
40+
class Solution:
41+
def findCheapestPrice(self, n: int, flights: List[List[int]], src: int, dst: int, K: int) -> int:
42+
43+
# construct graph
44+
graph_neighbor = collections.defaultdict(list)
45+
for s, e, p in flights:
46+
graph_neighbor[s].append((e, p))
47+
48+
# modified Dijkstra
49+
prices, steps = {}, {}
50+
min_heap = [(0, 0, src)]
51+
52+
while len(min_heap) > 0:
53+
price, step, node = heapq.heappop(min_heap)
54+
55+
if node == dst: # early return
56+
return price
57+
58+
if node not in prices:
59+
prices[node] = price
60+
61+
steps[node] = step
62+
if step <= K:
63+
step += 1
64+
for n, p in graph_neighbor[node]:
65+
if n not in prices or step < steps[n]:
66+
heapq.heappush(min_heap, (p + price, step, n))
67+
68+
return -1
69+
```

‎data_structure/heap.md

Lines changed: 48 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -171,9 +171,55 @@ class Solution:
171171
return ''.join(result)
172172
```
173173

174-
##Dijkstra's Algorithm
174+
### Prim's Algorithm
175175

176-
本质上也是 greedy + heap 的一种,用于求解图的单源最短路径相关的问题。
176+
实现上是 greedy + heap 的一个应用,用于构造图的最小生成树 (MST)。
177+
178+
### [minimum-risk-path](https://www.lintcode.com/problem/minimum-risk-path/description)
179+
180+
> 地图上有 m 条无向边,每条边 (x, y, w) 表示位置 m 到位置 y 的权值为 w。从位置 0 到 位置 n 可能有多条路径。我们定义一条路径的危险值为这条路径中所有的边的最大权值。请问从位置 0 到 位置 n 所有路径中最小的危险值为多少?
181+
182+
**图森面试真题**。最小危险值为最小生成树中 0 到 n 路径上的最大边权。
183+
184+
```Python
185+
class Solution:
186+
def getMinRiskValue(self, N, M, X, Y, W):
187+
188+
# construct graph
189+
adj = collections.defaultdict(list)
190+
for i in range(M):
191+
adj[X[i]].append((Y[i], W[i]))
192+
adj[Y[i]].append((X[i], W[i]))
193+
194+
# Prim's algorithm with min heap
195+
MST = collections.defaultdict(list)
196+
min_heap = [(w, 0, v) for v, w in adj[0]]
197+
heapq.heapify(min_heap)
198+
199+
while N not in MST:
200+
w, p, v = heapq.heappop(min_heap)
201+
if v not in MST:
202+
MST[p].append((v, w))
203+
MST[v].append((p, w))
204+
for n, w in adj[v]:
205+
if n not in MST:
206+
heapq.heappush(min_heap, (w, v, n))
207+
208+
# dfs to search route from 0 to n
209+
dfs = [(0, None, float('-inf'))]
210+
while dfs:
211+
v, p, max_w = dfs.pop()
212+
for n, w in MST[v]:
213+
cur_max_w = max(max_w, w)
214+
if n == N:
215+
return cur_max_w
216+
if n != p:
217+
dfs.append((n, v, cur_max_w))
218+
```
219+
220+
### Dijkstra's Algorithm
221+
222+
实现上是 greedy + heap 的一个应用,用于求解图的单源最短路径相关的问题。
177223

178224
### [network-delay-time](https://leetcode-cn.com/problems/network-delay-time/)
179225

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