The Tool Desk
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Choose an algorithm for the search you need
| Need | Input assumption | Useful structure | Important condition |
|---|---|---|---|
| Find an exact value or insertion boundary | Sequence already sorted by the search comparison rule | bisect_left or bisect_right |
Check the returned position if exact membership matters. |
| Test exact membership repeatedly | Arbitrary hashable values | set or dict |
Use a dictionary when values map to associated records. |
| Visit graph nodes by increasing number of edges from a start | Unweighted graph represented by neighbors | collections.deque |
Mark nodes discovered to prevent cycles and repeated queue entries. |
| Explore one branch deeply before backing up | Graph or state space | List as a LIFO stack | Track visited nodes when paths can loop. |
| Repeatedly select the smallest tentative distance or priority | Weighted graph or priority-driven state space | heapq |
Dijkstra requires nonnegative edge weights; use a tie-breaker for arbitrary payloads. |
These structures solve different problems. Binary search does not explore a graph, breadth-first search (BFS) does not minimize arbitrary weighted costs, and depth-first search (DFS) does not promise the shortest route. The right choice depends on both the input and the meaning of “found.”
Binary search and insertion boundaries with bisect
Python’s bisect module locates an insertion point in a sequence that is already ordered using the same comparison rule. It uses < to position values; it does not test equality to prove that a target exists. For exact matching, inspect the returned index and compare the item yourself. See the Python 3.14.7 bisect documentation.
Exact-match search, including an absent target
from bisect import bisect_left
def binary_search(items, target):
"""Return the first matching index, or -1 when target is absent.
items must already be sorted in ascending order under the same
comparison rule used for target.
"""
index = bisect_left(items, target)
if index != len(items) and items[index] == target:
return index
return -1
values = [2, 4, 4, 7, 10]
print(binary_search(values, 4)) # 1: first matching index
print(binary_search(values, 5)) # -1: absent
bisect_left returns the leftmost position at which the target can be inserted without disturbing the order. If duplicates exist, that position is the first equal value. The returned index can equal len(items), so guard against that before indexing.
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Inclusive and exclusive boundaries
The standard library handles the boundary logic without requiring a hand-written loop. bisect_right returns the position after all values equal to the target. Together, the two functions identify the half-open slice containing duplicates:
from bisect import bisect_left, bisect_right
values = [1, 3, 3, 3, 8]
left = bisect_left(values, 3)
right = bisect_right(values, 3)
print(left, right) # 1 4
print(values[left:right]) # [3, 3, 3]
Think of the result as a half-open interval [left, right): it includes index left and stops before index right. If left == right, there are no equal values, though that location remains a valid insertion boundary. This is useful for range queries such as all values within a lower and upper bound, where each endpoint’s inclusivity must be handled deliberately.
Sorted-list maintenance is not logarithmic
insort finds the insertion point with a bisection search, but inserting into a Python list shifts later elements. The search portion is O(log n); the list insertion is O(n) and dominates. Repeated insertion into a sorted list should therefore not be described as logarithmic-time insertion. For repeated exact lookups, Python’s documentation notes that dictionaries are more performant than bisection-based searching. The bisect functions are also not thread-safe if another thread concurrently uses or mutates the same sequence; coordinate access or use an appropriate synchronization strategy.
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Exact membership in an arbitrary collection
If values are not sorted and the question is simply “is this value present?”, sorting just to search once may add unnecessary work. A set is a direct representation for hashable values:
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blocked_ids = {18, 41, 77}
if 41 in blocked_ids:
print("already present")
Use a dictionary when a key identifies a record or value, for example user_by_id[user_id]. Sets and dictionaries are not substitutes for ordered range lookup: they do not tell you where a value belongs among neighboring values. For a workload that changes over time, account for the cost of maintaining the chosen structure, not only the cost of one lookup.
Breadth-first search with a FIFO queue
BFS visits a start node, then its neighbors, then nodes one edge farther away. In an unweighted graph this makes it suitable for finding a path with the fewest edges. Python’s tutorial demonstrates a FIFO queue using collections.deque, removing from the left with popleft() and appending generated moves. A visited/discovered set is essential for a general graph with cycles or multiple routes to a node.
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Runnable BFS path finder
from collections import deque
def bfs_path(graph, start, goal):
"""Return a shortest-by-edge-count path, or None if unreachable.
graph maps each node to an iterable of its neighboring nodes.
Nodes must be hashable.
"""
queue = deque([start])
parent = {start: None} # Also marks nodes as discovered.
while queue:
node = queue.popleft()
if node == goal:
path = []
while node is not None:
path.append(node)
node = parent[node]
return list(reversed(path))
for neighbor in graph.get(node, ()):
if neighbor not in parent:
parent[neighbor] = node
queue.append(neighbor)
return None
graph = {
"A": ["B", "C"],
"B": ["D"],
"C": ["D", "E"],
"D": ["F"],
"E": ["F"],
"F": [],
}
print(bfs_path(graph, "A", "F")) # ['A', 'B', 'D', 'F']
print(bfs_path(graph, "A", "Z")) # None
The implementation records a node’s parent when it is enqueued, rather than waiting until it is removed. That prevents the same node from being queued repeatedly through different routes. The parent map both serves as the discovered set and allows reconstruction of the path. If only reachability is needed, return a boolean instead.
Depth-first search with a stack
DFS uses a LIFO frontier: the most recently discovered node is explored next. It is useful for reachability, component exploration, and state-space traversal when breadth-first ordering is not needed. It does not guarantee a minimum-edge path. An explicit stack avoids Python recursion-depth limitations on deep or unbounded traversals.
def dfs_path(graph, start, goal):
"""Return one path if goal is reachable, otherwise None."""
stack = [start]
parent = {start: None}
while stack:
node = stack.pop()
if node == goal:
path = []
while node is not None:
path.append(node)
node = parent[node]
return list(reversed(path))
# Reverse to visit neighbors in their listed order with a LIFO stack.
for neighbor in reversed(list(graph.get(node, ()))):
if neighbor not in parent:
parent[neighbor] = node
stack.append(neighbor)
return None
As with BFS, marking a node when it is added to the frontier prevents cycles and duplicate work. The reverse is optional; it only makes traversal order align with the listed neighbor order. If neighbor order has no meaning, it can be omitted.
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Priority queues with heapq
heapq maintains a min-heap in a regular Python list: the smallest item is at index zero. heapify converts an existing list into a heap in linear time. When two entries have equal priorities, payload objects may not support ordering against one another; put a unique increasing counter between the priority and payload. These behaviors are documented in Python 3.14.7 heapq. The documentation reports explicit max-heap APIs as added in Python 3.14.
Stable priority entries with non-comparable tasks
import heapq
from itertools import count
sequence = count()
frontier = []
heapq.heappush(frontier, (5, next(sequence), {"job": "alpha"}))
heapq.heappush(frontier, (5, next(sequence), {"job": "beta"}))
heapq.heappush(frontier, (2, next(sequence), {"job": "urgent"}))
while frontier:
priority, _, task = heapq.heappop(frontier)
print(priority, task)
The second tuple field breaks ties without asking Python to compare the dictionaries. It also gives a consistent order among tasks with equal priorities. Without it, a tuple such as (priority, task) can raise TypeError when tied task objects are not orderable.
Dijkstra’s algorithm for nonnegative weighted edges
Dijkstra’s algorithm uses a min-heap to expand the currently cheapest known distance. It applies when edge weights are nonnegative. The example assumes a graph shaped like {node: [(neighbor, weight), ...]} and hashable node values:
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import heapq
from itertools import count
def dijkstra(graph, start):
"""Return shortest distances and predecessors from start.
All edge weights must be nonnegative.
"""
distances = {start: 0}
parent = {start: None}
sequence = count()
frontier = [(0, next(sequence), start)]
while frontier:
distance, _, node = heapq.heappop(frontier)
if distance != distances.get(node):
continue # Ignore a stale, more expensive heap entry.
for neighbor, weight in graph.get(node, ()):
if weight < 0:
raise ValueError("Dijkstra requires nonnegative edge weights")
candidate = distance + weight
if candidate < distances.get(neighbor, float("inf")):
distances[neighbor] = candidate
parent[neighbor] = node
heapq.heappush(
frontier, (candidate, next(sequence), neighbor)
)
return distances, parent
def reconstruct_path(parent, goal):
if goal not in parent:
return None
path = []
node = goal
while node is not None:
path.append(node)
node = parent[node]
return list(reversed(path))
weighted = {
"A": [("B", 4), ("C", 1)],
"B": [("D", 1)],
"C": [("B", 2), ("D", 7)],
"D": [],
}
distances, parent = dijkstra(weighted, "A")
print(distances["D"]) # 4
print(reconstruct_path(parent, "D")) # ['A', 'C', 'B', 'D']
The heap may contain an older entry for a node after a shorter route is discovered; comparing the popped distance against the current best distance discards that stale entry. Distances are returned only for nodes reachable from the start. A negative edge invalidates Dijkstra's assumptions, so choose an algorithm designed for the graph's weight rules instead. Runtime depends on graph representation and heap operations; do not apply one complexity figure without stating those assumptions.
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- Sorted records by a key: keep the sequence sorted under the same key/comparison rule used in lookup. For complex records, use the documented
keysupport inbisectcarefully and ensure the searched value is in the expected key domain. - Multiple equivalent values: use left and right boundaries to retrieve a duplicate range; validate exact equality separately when checking membership.
- Directed graphs: include only outgoing neighbors in the adjacency list. If you need reverse reachability, represent or construct reverse edges explicitly.
- State-space search: make states immutable/hashable or derive a stable key for the discovered set. If the state omits relevant information, two distinct situations may be incorrectly treated as the same state.
- Need a weighted minimum path: use a priority-driven algorithm with assumptions appropriate to the edge costs. Do not substitute BFS unless all steps have equal cost.
Common errors and fixes
| Symptom | Likely cause | Fix |
|---|---|---|
| Binary search misses a value that appears in the input | The input is unsorted, or sorted with a different key/comparison rule. | Sort or maintain the sequence under the same ordering before calling bisect. |
IndexError after bisect_left |
The target belongs after the final item, so the insertion point is len(items). |
Check the index against the sequence length before indexing. |
| A lookup reports a match for a value that is not present | The code treats the insertion point as proof of membership. | Compare items[index] == target after confirming the index is in range. |
| BFS or DFS loops or grows without bound | Nodes are not marked discovered, or the state key does not represent the full state. | Record each node/state when adding it to the frontier; revisit the state model if needed. |
Heap raises TypeError for tied priorities |
Tuple comparison reaches non-comparable task values. | Use a unique counter as a tie-breaker between priority and payload. |
| Dijkstra returns an implausible result with negative edges | The algorithm's nonnegative-weight precondition is violated. | Use a shortest-path method that supports negative weights, or correct the input if weights should be nonnegative. |
| Results change unpredictably during bisection | Another thread is using or mutating the same sequence concurrently. | Protect access with synchronization or search an immutable/safely isolated sequence. |
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Does bisect confirm that an item exists?
No. It returns an insertion point; exact-match code must check that the index is in range and the indexed value equals the target.
Can breadth-first search find the cheapest path in a weighted graph?
Not in general. BFS minimizes edge count in an unweighted graph; use an algorithm whose assumptions match the edge weights, such as Dijkstra for nonnegative weights.
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