diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml
index 39daf3dd7f88..b865f88350ea 100644
--- a/.pre-commit-config.yaml
+++ b/.pre-commit-config.yaml
@@ -19,7 +19,7 @@ repos:
       - id: auto-walrus
 
   - repo: https://github.com/astral-sh/ruff-pre-commit
-    rev: v0.15.9
+    rev: v0.15.15
     hooks:
       - id: ruff-check
       - id: ruff-format
@@ -32,7 +32,7 @@ repos:
           - tomli
 
   - repo: https://github.com/tox-dev/pyproject-fmt
-    rev: v2.21.0
+    rev: v2.23.0
     hooks:
       - id: pyproject-fmt
 
diff --git a/DIRECTORY.md b/DIRECTORY.md
index ca454bd5fd82..daf71bab8162 100644
--- a/DIRECTORY.md
+++ b/DIRECTORY.md
@@ -471,6 +471,8 @@
   * [Geometry](geometry/geometry.py)
   * [Graham Scan](geometry/graham_scan.py)
   * [Jarvis March](geometry/jarvis_march.py)
+  * [Ramer Douglas Peucker](geometry/ramer_douglas_peucker.py)
+  * [Segment Intersection](geometry/segment_intersection.py)
   * Tests
     * [Test Graham Scan](geometry/tests/test_graham_scan.py)
     * [Test Jarvis March](geometry/tests/test_jarvis_march.py)
@@ -523,6 +525,7 @@
   * [Graphs Floyd Warshall](graphs/graphs_floyd_warshall.py)
   * [Greedy Best First](graphs/greedy_best_first.py)
   * [Greedy Min Vertex Cover](graphs/greedy_min_vertex_cover.py)
+  * [Johnson](graphs/johnson.py)
   * [Kahns Algorithm Long](graphs/kahns_algorithm_long.py)
   * [Kahns Algorithm Topo](graphs/kahns_algorithm_topo.py)
   * [Karger](graphs/karger.py)
@@ -543,6 +546,7 @@
   * [Strongly Connected Components](graphs/strongly_connected_components.py)
   * [Tarjans Scc](graphs/tarjans_scc.py)
   * Tests
+    * [Test Johnson](graphs/tests/test_johnson.py)
     * [Test Min Spanning Tree Kruskal](graphs/tests/test_min_spanning_tree_kruskal.py)
     * [Test Min Spanning Tree Prim](graphs/tests/test_min_spanning_tree_prim.py)
 
diff --git a/geometry/ramer_douglas_peucker.py b/geometry/ramer_douglas_peucker.py
new file mode 100644
index 000000000000..a03bbb2e5086
--- /dev/null
+++ b/geometry/ramer_douglas_peucker.py
@@ -0,0 +1,184 @@
+"""
+Ramer-Douglas-Peucker polyline simplification algorithm.
+
+Given a sequence of 2-D points and a tolerance epsilon, the algorithm
+reduces the number of points while preserving the overall shape of the curve.
+
+Time complexity:  O(n log n) average, O(n²) worst case
+Space complexity: O(n)
+
+References:
+    https://en.wikipedia.org/wiki/Ramer%E2%80%93Douglas%E2%80%93Peucker_algorithm
+"""
+
+from __future__ import annotations
+
+import math
+
+
+def _euclidean_distance(
+    point_a: tuple[float, float],
+    point_b: tuple[float, float],
+) -> float:
+    """Return the Euclidean distance between two 2-D points.
+
+    >>> _euclidean_distance((0.0, 0.0), (3.0, 4.0))
+    5.0
+    >>> _euclidean_distance((1.0, 1.0), (1.0, 1.0))
+    0.0
+    """
+    return math.hypot(point_b[0] - point_a[0], point_b[1] - point_a[1])
+
+
+def _perpendicular_distance(
+    point: tuple[float, float],
+    line_start: tuple[float, float],
+    line_end: tuple[float, float],
+) -> float:
+    """Return the distance from *point* to the line **segment** between
+    *line_start* and *line_end*.
+
+    When the perpendicular projection of *point* onto the infinite line falls
+    within the segment, this equals the perpendicular distance to that line.
+    When the projection falls outside the segment, the distance to the nearest
+    endpoint is returned instead (projection clamped to [0, 1]).
+
+    This is the correct distance measure for the Ramer-Douglas-Peucker
+    algorithm: using the infinite-line distance can incorrectly discard points
+    whose projection lies beyond a segment endpoint.
+
+    >>> _perpendicular_distance((4.0, 0.0), (0.0, 0.0), (0.0, 3.0))
+    4.0
+    >>> # order of line_start and line_end does not affect the result
+    >>> _perpendicular_distance((4.0, 0.0), (0.0, 3.0), (0.0, 0.0))
+    4.0
+    >>> _perpendicular_distance((4.0, 1.0), (0.0, 1.0), (0.0, 4.0))
+    4.0
+    >>> _perpendicular_distance((2.0, 1.0), (-2.0, 1.0), (-2.0, 4.0))
+    4.0
+    >>> # projection falls outside the segment; distance to nearest endpoint
+    >>> round(_perpendicular_distance((0.0, 2.0), (1.0, 0.0), (3.0, 0.0)), 6)
+    2.236068
+    """
+    px, py = point
+    ax, ay = line_start
+    bx, by = line_end
+    dx, dy = bx - ax, by - ay
+    seg_len_sq = dx * dx + dy * dy
+    if seg_len_sq == 0.0:
+        # line_start and line_end coincide; fall back to point-to-point distance
+        return _euclidean_distance(point, line_start)
+    # Project point onto the segment line, then clamp t to [0, 1] so the
+    # nearest point is always on the segment rather than the infinite line.
+    t = max(0.0, min(1.0, ((px - ax) * dx + (py - ay) * dy) / seg_len_sq))
+    nearest_x = ax + t * dx
+    nearest_y = ay + t * dy
+    return math.hypot(px - nearest_x, py - nearest_y)
+
+
+def ramer_douglas_peucker(
+    pts: list[tuple[float, float]],
+    epsilon: float,
+) -> list[tuple[float, float]]:
+    """Simplify a polyline using the Ramer-Douglas-Peucker algorithm.
+
+    Given a sequence of 2-D points and a maximum allowable deviation
+    *epsilon* (>= 0), returns a simplified list of points such that no
+    discarded point is farther than *epsilon* from the simplified polyline.
+
+    Parameters
+    ----------
+    pts:
+        Ordered sequence of ``(x, y)`` points describing the polyline.
+    epsilon:
+        Maximum allowable distance of any discarded point from the
+        simplified polyline.  Must be non-negative.
+
+    Returns
+    -------
+    list[tuple[float, float]]
+        Simplified list of ``(x, y)`` points.  The first and last points of
+        *pts* are always preserved.
+
+    Raises
+    ------
+    ValueError
+        If *epsilon* is negative.
+
+    References
+    ----------
+    https://en.wikipedia.org/wiki/Ramer%E2%80%93Douglas%E2%80%93Peucker_algorithm
+
+    Examples
+    --------
+    >>> ramer_douglas_peucker([], epsilon=1.0)
+    []
+    >>> ramer_douglas_peucker([(0.0, 0.0)], epsilon=1.0)
+    [(0.0, 0.0)]
+    >>> ramer_douglas_peucker([(0.0, 0.0), (1.0, 0.0)], epsilon=1.0)
+    [(0.0, 0.0), (1.0, 0.0)]
+    >>> # middle point is within epsilon - it is discarded
+    >>> ramer_douglas_peucker([(0.0, 0.0), (1.0, 0.1), (2.0, 0.0)], epsilon=0.5)
+    [(0.0, 0.0), (2.0, 0.0)]
+    >>> # middle point exceeds epsilon - it is kept
+    >>> ramer_douglas_peucker([(0.0, 0.0), (1.0, 1.0), (2.0, 0.0)], epsilon=0.5)
+    [(0.0, 0.0), (1.0, 1.0), (2.0, 0.0)]
+    >>> ramer_douglas_peucker([(0.0, 0.0), (1.0, 0.5), (2.0, 0.0)], epsilon=-1.0)
+    Traceback (most recent call last):
+        ...
+    ValueError: epsilon must be non-negative, got -1.0
+    """
+    if epsilon < 0:
+        msg = f"epsilon must be non-negative, got {epsilon!r}"
+        raise ValueError(msg)
+
+    if len(pts) < 3:
+        return list(pts)
+
+    # ---------------------------------------------------------------------------
+    # Iterative, stack-based implementation.
+    #
+    # The naive recursive approach copies sublists at every level via slicing
+    # (pts[:max_index+1] / pts[max_index:]), which is O(n) per call and makes
+    # the overall algorithm O(n²) in memory even for well-balanced splits.  An
+    # explicit stack operating on index ranges avoids all copying and also
+    # eliminates the risk of hitting Python's recursion limit for long polylines.
+    # ---------------------------------------------------------------------------
+    n = len(pts)
+
+    # keep[i] is True when pts[i] must appear in the output.
+    keep: list[bool] = [False] * n
+    keep[0] = True
+    keep[-1] = True
+
+    # Stack of (start_index, end_index) pairs still to be examined.
+    stack: list[tuple[int, int]] = [(0, n - 1)]
+
+    while stack:
+        start, end = stack.pop()
+        if end - start < 2:
+            # Only one interior candidate at most; nothing to split further.
+            continue
+
+        # Find the interior point with the greatest distance to the segment.
+        max_dist = 0.0
+        max_index = start
+        for i in range(start + 1, end):
+            dist = _perpendicular_distance(pts[i], pts[start], pts[end])
+            if dist > max_dist:
+                max_dist = dist
+                max_index = i
+
+        if max_dist > epsilon:
+            keep[max_index] = True
+            stack.append((start, max_index))
+            stack.append((max_index, end))
+        # else: all interior points are within epsilon; discard them all.
+
+    return [pts[i] for i in range(n) if keep[i]]
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
diff --git a/geometry/segment_intersection.py b/geometry/segment_intersection.py
new file mode 100644
index 000000000000..e2e2e10f1e4d
--- /dev/null
+++ b/geometry/segment_intersection.py
@@ -0,0 +1,112 @@
+"""
+Given two line segments, determine whether they intersect.
+
+This is based on the algorithm described in Introduction to Algorithms
+(CLRS), Chapter 33.
+
+Reference:
+    - https://en.wikipedia.org/wiki/Line%E2%80%93line_intersection
+    - https://en.wikipedia.org/wiki/Orientation_(geometry)
+"""
+
+from __future__ import annotations
+
+from typing import NamedTuple
+
+
+class Point(NamedTuple):
+    """A point in 2D space.
+
+    >>> Point(0, 0)
+    Point(x=0, y=0)
+    >>> Point(1, -3)
+    Point(x=1, y=-3)
+    """
+
+    x: float
+    y: float
+
+
+def direction(pivot: Point, target: Point, query: Point) -> float:
+    """Return the cross product of vectors (pivot->query) and (pivot->target).
+
+    The sign of the result encodes the orientation of the ordered triple
+    (pivot, target, query):
+      - Negative  ->  counter-clockwise (left turn)
+      - Positive  ->  clockwise (right turn)
+      - Zero      ->  collinear
+
+    >>> direction(Point(0, 0), Point(1, 0), Point(0, 1))
+    -1
+    >>> direction(Point(0, 0), Point(0, 1), Point(1, 0))
+    1
+    >>> direction(Point(0, 0), Point(1, 1), Point(2, 2))
+    0
+    """
+    return (query.x - pivot.x) * (target.y - pivot.y) - (target.x - pivot.x) * (
+        query.y - pivot.y
+    )
+
+
+def on_segment(seg_start: Point, seg_end: Point, point: Point) -> bool:
+    """Check whether *point*, known to be collinear with the segment, lies on it.
+
+    >>> on_segment(Point(0, 0), Point(4, 4), Point(2, 2))
+    True
+    >>> on_segment(Point(0, 0), Point(4, 4), Point(5, 5))
+    False
+    >>> on_segment(Point(0, 0), Point(4, 0), Point(2, 0))
+    True
+    """
+    return min(seg_start.x, seg_end.x) <= point.x <= max(
+        seg_start.x, seg_end.x
+    ) and min(seg_start.y, seg_end.y) <= point.y <= max(seg_start.y, seg_end.y)
+
+
+def segments_intersect(p1: Point, p2: Point, p3: Point, p4: Point) -> bool:
+    """Return True if line segment p1p2 intersects line segment p3p4.
+
+    Uses the CLRS cross-product / orientation method.  Handles both the
+    general case (proper crossing) and degenerate cases where one endpoint
+    lies exactly on the other segment.
+
+    >>> segments_intersect(Point(0, 0), Point(2, 2), Point(0, 2), Point(2, 0))
+    True
+    >>> segments_intersect(Point(0, 0), Point(2, 2), Point(1, 1), Point(3, 3))
+    True
+    >>> segments_intersect(Point(0, 0), Point(1, 0), Point(2, 0), Point(3, 0))
+    False
+    >>> segments_intersect(Point(0, 0), Point(1, 1), Point(1, 0), Point(2, 1))
+    False
+    >>> segments_intersect(Point(0, 0), Point(1, 1), Point(0, 1), Point(0, 2))
+    False
+    >>> segments_intersect(Point(0, 0), Point(1, 0), Point(1, 0), Point(2, 0))
+    True
+    """
+    d1 = direction(p3, p4, p1)
+    d2 = direction(p3, p4, p2)
+    d3 = direction(p1, p2, p3)
+    d4 = direction(p1, p2, p4)
+
+    if ((d1 < 0 < d2) or (d2 < 0 < d1)) and ((d3 < 0 < d4) or (d4 < 0 < d3)):
+        return True
+
+    if d1 == 0 and on_segment(p3, p4, p1):
+        return True
+    if d2 == 0 and on_segment(p3, p4, p2):
+        return True
+    if d3 == 0 and on_segment(p1, p2, p3):
+        return True
+    return d4 == 0 and on_segment(p1, p2, p4)
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    print("Enter four points as 'x y' pairs (one per line):")
+    points = [Point(*map(float, input().split())) for _ in range(4)]
+    p1, p2, p3, p4 = points
+    result = segments_intersect(p1, p2, p3, p4)
+    print(1 if result else 0)
diff --git a/graphs/johnson.py b/graphs/johnson.py
new file mode 100644
index 000000000000..6306ab5f8654
--- /dev/null
+++ b/graphs/johnson.py
@@ -0,0 +1,118 @@
+import heapq
+from collections.abc import Hashable
+
+Node = Hashable
+edge = tuple[Node, Node, float]
+adjacency = dict[Node, list[tuple[Node, float]]]
+
+
+def _collect_nodes_and_edges(graph: adjacency) -> tuple[list[Node], list[edge]]:
+    nodes = set()
+    edges: list[edge] = []
+    for u, neighbors in graph.items():
+        nodes.add(u)
+        for v, w in neighbors:
+            nodes.add(v)
+            edges.append((u, v, w))
+    return list(nodes), edges
+
+
+def _bellman_ford(nodes: list[Node], edges: list[edge]) -> dict[Node, float]:
+    """
+    Bellman-Ford relaxation to compute potentials h[v] for all vertices.
+    Raises ValueError if a negative weight cycle exists.
+    """
+    dist: dict[Node, float] = dict.fromkeys(nodes, 0.0)
+    n = len(nodes)
+
+    for _ in range(n - 1):
+        updated = False
+        for u, v, w in edges:
+            if dist[u] + w < dist[v]:
+                dist[v] = dist[u] + w
+                updated = True
+        if not updated:
+            break
+    else:
+        for u, v, w in edges:
+            if dist[u] + w < dist[v]:
+                raise ValueError("Negative weight cycle detected")
+    return dist
+
+
+def _dijkstra(
+    start: Node,
+    nodes: list[Node],
+    graph: adjacency,
+    potentials: dict[Node, float],
+) -> dict[Node, float]:
+    """
+    Dijkstra over reweighted graph, using potentials h to make weights non-negative.
+    Returns distances from start in the reweighted space.
+    """
+    inf = float("inf")
+    dist: dict[Node, float] = dict.fromkeys(nodes, inf)
+    dist[start] = 0.0
+    heap: list[tuple[float, Node]] = [(0.0, start)]
+
+    while heap:
+        d_u, u = heapq.heappop(heap)
+        if d_u > dist[u]:
+            continue
+        for v, w in graph.get(u, []):
+            w_prime = w + potentials[u] - potentials[v]
+            if w_prime < 0:
+                raise ValueError(
+                    "Negative edge weight after reweighting: numeric error"
+                )
+            new_dist = d_u + w_prime
+            if new_dist < dist[v]:
+                dist[v] = new_dist
+                heapq.heappush(heap, (new_dist, v))
+    return dist
+
+
+def johnson(graph: adjacency) -> dict[Node, dict[Node, float]]:
+    """
+    Compute all-pairs shortest paths using Johnson's algorithm.
+
+    Reference:
+        https://en.wikipedia.org/wiki/Johnson%27s_algorithm
+
+    Args:
+        graph: adjacency list {u: [(v, weight), ...], ...}
+
+    Returns:
+        dict of dicts: dist[u][v] = shortest distance from u to v
+
+    Raises:
+        ValueError: if a negative weight cycle is detected
+
+    Example:
+    >>> g = {
+    ...     0: [(1, 3), (2, 8), (4, -4)],
+    ...     1: [(3, 1), (4, 7)],
+    ...     2: [(1, 4)],
+    ...     3: [(0, 2), (2, -5)],
+    ...     4: [(3, 6)],
+    ... }
+    >>> round(johnson(g)[0][3], 2)
+    2.0
+    """
+    nodes, edges = _collect_nodes_and_edges(graph)
+    potentials = _bellman_ford(nodes, edges)
+
+    all_pairs: dict[Node, dict[Node, float]] = {}
+    inf = float("inf")
+    for s in nodes:
+        dist_reweighted = _dijkstra(s, nodes, graph, potentials)
+        dists_orig: dict[Node, float] = {}
+        for v in nodes:
+            d_prime = dist_reweighted[v]
+            if d_prime < inf:
+                dists_orig[v] = d_prime - potentials[s] + potentials[v]
+            else:
+                dists_orig[v] = inf
+        all_pairs[s] = dists_orig
+
+    return all_pairs
diff --git a/graphs/tests/test_johnson.py b/graphs/tests/test_johnson.py
new file mode 100644
index 000000000000..e149aac85d0f
--- /dev/null
+++ b/graphs/tests/test_johnson.py
@@ -0,0 +1,24 @@
+import math
+
+import pytest
+
+from graphs.johnson import johnson
+
+
+def test_johnson_basic():
+    g = {
+        0: [(1, 3), (2, 8), (4, -4)],
+        1: [(3, 1), (4, 7)],
+        2: [(1, 4)],
+        3: [(0, 2), (2, -5)],
+        4: [(3, 6)],
+    }
+    dist = johnson(g)
+    assert math.isclose(dist[0][3], 2.0, abs_tol=1e-9)
+    assert math.isclose(dist[3][2], -5.0, abs_tol=1e-9)
+
+
+def test_johnson_negative_cycle():
+    g2 = {0: [(1, 1)], 1: [(0, -3)]}
+    with pytest.raises(ValueError):
+        johnson(g2)
diff --git a/pyproject.toml b/pyproject.toml
index 34e099a46435..0b0c3f5cfda4 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -169,14 +169,14 @@ skip = """\
 python_version = "3.14"
 
 [tool.pytest]
-ini_options.markers = [
-  "mat_ops: mark a test as utilizing matrix operations.",
-]
 ini_options.addopts = [
   "--durations=10",
   "--doctest-modules",
   "--showlocals",
 ]
+ini_options.markers = [
+  "mat_ops: mark a test as utilizing matrix operations.",
+]
 
 [tool.coverage]
 report.omit = [
diff --git a/sorts/tim_sort.py b/sorts/tim_sort.py
index 41ab4a10a87b..2eeed88b7399 100644
--- a/sorts/tim_sort.py
+++ b/sorts/tim_sort.py
@@ -1,4 +1,7 @@
-def binary_search(lst, item, start, end):
+from typing import Any
+
+
+def binary_search(lst: list[Any], item: Any, start: int, end: int) -> int:
     if start == end:
         return start if lst[start] > item else start + 1
     if start > end:
@@ -13,7 +16,7 @@ def binary_search(lst, item, start, end):
         return mid
 
 
-def insertion_sort(lst):
+def insertion_sort(lst: list[Any]) -> list[Any]:
     length = len(lst)
 
     for index in range(1, length):
@@ -24,7 +27,7 @@ def insertion_sort(lst):
     return lst
 
 
-def merge(left, right):
+def merge(left: list[Any], right: list[Any]) -> list[Any]:
     if not left:
         return right
 
@@ -37,7 +40,7 @@ def merge(left, right):
     return [right[0], *merge(left, right[1:])]
 
 
-def tim_sort(lst):
+def tim_sort(lst: list[Any] | tuple[Any, ...] | str) -> list[Any]:
     """
     >>> tim_sort("Python")
     ['P', 'h', 'n', 'o', 't', 'y']
@@ -53,7 +56,7 @@ def tim_sort(lst):
     length = len(lst)
     runs, sorted_runs = [], []
     new_run = [lst[0]]
-    sorted_array = []
+    sorted_array: list[Any] = []
     i = 1
     while i < length:
         if lst[i] < lst[i - 1]: