ManhattanDistance¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

ManhattanDistance[u, v]

Gives Sum Abs[u_i - v_i], the sum of component-wise distances. Differs from EuclideanDistance in two or more dimensions; in one dimension the two agree.

Examples (8)¶

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (6)¶

In[1]:= EuclideanDistance[{1, 2}, {4, 6}]
Out[1]= 5

In[2]:= SquaredEuclideanDistance[{1/3, 0}, {0, 1/7}]
Out[2]= 58/441

In[3]:= ManhattanDistance[{1, 2}, {4, 6}]
Out[3]= 7

In[4]:= EuclideanDistance[{0, 0}, {1, 1}]
Out[4]= Sqrt[2]

In[5]:= CosineDistance[{1, 0}, {0, 1}]
Out[5]= 1

In[6]:= CosineDistance[{1, 0}, {-1, 0}]
Out[6]= 2

Applications (2)¶

Sum of coordinate gaps: 3 + 4 = 7

In[7]:= ManhattanDistance[{0, 0}, {3, 4}]
Out[7]= 7

A symbolic pair stays symbolic

In[8]:= ManhattanDistance[{a}, {b}]
Out[8]= Abs[a - b]

Implementation notes¶

Algorithm. builtin_manhattan_distance computes Sum_i Abs[u_i - v_i] via the shared dist_builtin(res, p=1, root=false). dist_sum forms the sum of absolute component differences by composing internal_subtract, the dist_abs helper and internal_plus through eval_and_free, so exact input stays exact (ManhattanDistance[{1, 2}, {4, 6}] is 7), complex components contribute their modulus, and a symbolic pair comes back symbolically (ManhattanDistance[{a}, {b}] is Abs[a - b], as Mathematica answers).

Shape. dist_shape admits two scalars or two equal-length Lists only; a length mismatch, a list-valued component, or a non-atomic "scalar" (a Rational is stored as Rational[...], an EXPR_FUNCTION, so a bare rational scalar is not accepted) declines. dist_abs decides a real sign with list_numeric_sign to route around the Abs bigint-rational gap.

Complexity / limits. O(n) arithmetic evaluations over the components; interpreter-speed. ATTR_PROTECTED. See EuclideanDistance, SquaredEuclideanDistance, and CosineDistance — all share the dist_sum loop and differ only in p and whether a root is taken.

Protected. Not Listable: threading over a List argument is exactly what these must not do, because the list is the point.
Exact input gives an exact result where the value is rational. SquaredEuclideanDistance[{1, 2}, {4, 6}] is 25, not 25., and SquaredEuclideanDistance[{1/3, 0}, {0, 1/7}] is 58/441. Squared Euclidean is monotone in Euclidean, so ranking on it orders points identically without introducing a root -- which is how FindClusters stays exact in n dimensions.
Complex components contribute their modulus, because the definition takes Abs before squaring rather than squaring the difference. This matters only for complex input, where the two orders differ, and follows Mathematica.
Symbolic input survives rather than being rejected: ManhattanDistance[{a}, {b}] is Abs[a - b], as in Mathematica.
CosineDistance ranges over [0, 2] -- 0 parallel, 1 orthogonal, 2 antiparallel -- and ignores magnitude. It is not a metric (it violates the triangle inequality) and has no squared form that ranks identically, so it is used directly. A zero vector on either side gives 0, following Mathematica; that is a convention, not a derivation, since the quotient is 0/0.
Mismatched lengths, or an argument that is a matrix, leave the call unevaluated.

Attributes: Protected.

References¶

Source: src/list/distance.c
Specification: docs/spec/builtins/lists-and-iteration.md
Tests: tests/test_list.c

Notes & additional examples¶

Notes¶

ManhattanDistance[u, v] is Sum Abs[u_i - v_i], the taxicab (city-block) distance — the distance travelled along axis-aligned moves rather than in a straight line. Both arguments must be scalars or lists of equal length.

Exact input stays exact, complex components contribute their modulus, and a symbolic pair returns Abs[a - b] rather than an error, exactly as Mathematica answers. Note that a bare rational scalar such as ManhattanDistance[1/2, 3/2] is left unevaluated, because a Rational is a compound expression rather than an atomic scalar — wrap the values in lists (ManhattanDistance[{1/2}, {3/2}]) to measure them.