pythonの幅優先探索について教えてください

前提・実現したいこと

閲覧ありがとうございます。
現在Pythonでプログラミングを学んでいるのですが幅優先探索で躓いてます。
下記のようなグラフを作り具体例のような処理を行いたいのですが上手く動作させることが出来ません。
どなたかご教授していただけると助かります。

具体例

 1.Starting node（1000）の隣接ノードa,b,cの中で一番重みの大きいaに100移す。
 2.Starting nodeの隣接ノード（a以外）で重みが一番小さいbに500移す。
 3.Starting nodeの隣接ノード（a,b以外）で重みが一番小さいcに400移す。
 4.aの隣接ノードの中で一番重みの小さいdに100移す。
 5.bの隣接ノードで重みが一番小さいeに350移す。
 6.bの隣接ノード（e以外）で重みが一番小さいfに150移す。
 7.cの隣接ノードの中で一番重みの小さいgに400移す。
 8.dの隣接ノードの中で一番重みの小さいhに100移す。
 9.gの隣接ノードの中で一番重みの小さいiに400移す。
10.hの隣接ノードの中で一番重みの小さいEnd nodeに100移す。
11.eの隣接ノードで重みが一番小さいEnd nodeに350移す。
12.fの隣接ノードで重みが一番小さいEnd nodeに150移す。
13.iの隣接ノードで重みが一番小さいEnd nodeに400移す。

ソースコード

python
1#graph
2graph = {}
3graph["start"] = {}
4graph["start"]["a"] = 1
5graph["start"]["b"] = 3
6graph["start"]["c"] = 4
7
8graph["a"] = {}
9graph["a"]["d"] = 5
10graph["d"] = {}
11graph["d"]["h"] = 7
12graph["h"] = {}
13graph["h"]["end"] = 2
14
15graph["b"] = {}
16graph["b"]["e"] = 1
17graph["b"]["f"] = 2
18graph["e"] = {}
19graph["e"]["end"] = 3
20graph["f"] = {}
21graph["f"]["end"] = 7
22
23graph["c"] = {}
24graph["c"]["g"] = 4
25graph["g"] = {}
26graph["g"]["i"] = 10
27graph["i"] = {}
28graph["i"]["end"] = 1
29
30graph["end"] = {}
31
32#cost
33infinity = float("inf")
34costs = {}
35costs["a"] = 100
36costs["d"] = 100
37costs["h"] = 100
38
39costs["b"] = 500
40costs["e"] = 350
41costs["f"] = 150
42
43costs["c"] = 400
44costs["g"] = 400
45costs["i"] = 400
46costs["end"] = infinity
47
48# the parents table
49parents = {}
50parents["a"] = "start"
51parents["d"] = "a"
52parents["h"] = "d"
53
54parents["b"] = "start"
55parents["e"] = "b"
56parents["f"] = "b"
57
58parents["c"] = "start"
59parents["g"] = "c"
60parents["i"] = "g"
61
62parents["end"] = None
63
64processed = []
65
66def find_lowest_cost_node(costs):
67    lowest_cost = float("end")
68    lowest_cost_node = None
69    # Go through each node.
70    for node in costs:
71        cost = costs[node]
72        # If it's the lowest cost so far and hasn't been processed yet...
73        if cost < lowest_cost and node not in processed:
74            # ... set it as the new lowest-cost node.
75            lowest_cost = cost
76            lowest_cost_node = node
77    return lowest_cost_node
78
79# Find the lowest-cost node that you haven't processed yet.
80node = find_lowest_cost_node(costs)
81# If you've processed all the nodes, this while loop is done.
82while node is not None:
83    cost = costs[node]
84    # Go through all the neighbors of this node.
85    neighbors = graph[node]
86    for n in neighbors.keys():
87        new_cost = cost + neighbors[n]
88        # If it's cheaper to get to this neighbor by going through this node...
89        if costs[n] > new_cost:
90            # ... update the cost for this node.
91            costs[n] = new_cost
92            # This node becomes the new parent for this neighbor.
93            parents[n] = node
94    # Mark the node as processed.
95    processed.append(node)
96    # Find the next node to process, and loop.
97    node = find_lowest_cost_node(costs)
98
99print "Cost from the start to each node:"
100print costs

発生している問題・エラーメッセージ

Traceback (most recent call last):
  File "sample.py", line 76, in <module>
    node = find_lowest_cost_node(costs)
  File "sample.py", line 63, in find_lowest_cost_node
    lowest_cost = float("end")
ValueError: could not convert string to float: end