Natural Language Processing · TEE · formulas

Manhattan, Euclidean and Minkowski: three complete comparisons

Representation & similarity

Formulas explained through full worked examples

Distance measures separation; smaller means closer. Absolute values prevent positive and negative differences from cancelling.

d_1=\sum_i|x_i-y_i|\quad\text{Manhattan}

d_2=\sqrt{\sum_i(x_i-y_i)^2}\quad\text{Euclidean}

d_p=\left(\sum_i|x_i-y_i|^p\right)^{1/p},\quad p\ge1\quad\text{Minkowski}.

Manhattan is p=1; Euclidean is p=2. For p=3, cube absolute differences, add, then take a cube root.

Question. Find Manhattan, Euclidean and Minkowski p=3 distances for x=[1,2], y=[4,6].

Given

Vectors share coordinate order; negative inputs are allowed.

To find

Find Manhattan, Euclidean and Minkowski p=3 distances for x=[1,2], y=[4,6].

Formula and why it applies

Use $d_1=\sum|\Delta_i|$ , $d_2=\sqrt{\sum|\Delta_i|^2}$ , $d_3=\sqrt[3]{\sum|\Delta_i|^3}$ .

Step 1. Absolute differences

|1 − (4)| = 3; |2 − (6)| = 4.

Step 2. Manhattan

3 + 4 = 7.

Step 3. Square and add

3²=9; 4²=16. Sum=25.

Step 4. Euclidean root

$\sqrt{25}\approx 5.000000.$

Step 5. Cube and add

3³=27; 4³=64. Sum=91.

Step 6. Minkowski root

$\sqrt[3]{91}\approx 4.497941.$

Answer. Manhattan=7; Euclidean≈5.000000; p=3≈4.497941.

Question. Find Manhattan, Euclidean and Minkowski p=3 distances for x=[0,1,2], y=[2,1,0].

Given

Vectors share coordinate order; negative inputs are allowed.

To find

Find Manhattan, Euclidean and Minkowski p=3 distances for x=[0,1,2], y=[2,1,0].

Formula and why it applies

Use $d_1=\sum|\Delta_i|$ , $d_2=\sqrt{\sum|\Delta_i|^2}$ , $d_3=\sqrt[3]{\sum|\Delta_i|^3}$ .

Step 1. Absolute differences

|0 − (2)| = 2; |1 − (1)| = 0; |2 − (0)| = 2.

Step 2. Manhattan

2 + 0 + 2 = 4.

Step 3. Square and add

2²=4; 0²=0; 2²=4. Sum=8.

Step 4. Euclidean root

$\sqrt{8}\approx 2.828427.$

Step 5. Cube and add

2³=8; 0³=0; 2³=8. Sum=16.

Step 6. Minkowski root

$\sqrt[3]{16}\approx 2.519842.$

Answer. Manhattan=4; Euclidean≈2.828427; p=3≈2.519842.

Question. Find Manhattan, Euclidean and Minkowski p=3 distances for x=[-1,3], y=[2,-1].

Given

Vectors share coordinate order; negative inputs are allowed.

To find

Find Manhattan, Euclidean and Minkowski p=3 distances for x=[-1,3], y=[2,-1].

Formula and why it applies

Use $d_1=\sum|\Delta_i|$ , $d_2=\sqrt{\sum|\Delta_i|^2}$ , $d_3=\sqrt[3]{\sum|\Delta_i|^3}$ .

Step 1. Absolute differences

|-1 − (2)| = 3; |3 − (-1)| = 4.

Step 2. Manhattan

3 + 4 = 7.

Step 3. Square and add

3²=9; 4²=16. Sum=25.

Step 4. Euclidean root

$\sqrt{25}\approx 5.000000.$

Step 5. Cube and add

3³=27; 4³=64. Sum=91.

Step 6. Minkowski root

$\sqrt[3]{91}\approx 4.497941.$

Answer. Manhattan=7; Euclidean≈5.000000; p=3≈4.497941.

Manhattan: absolute-add. Euclidean: square-add-root. Minkowski: p-power-add-p-root.