Assign and Update

When Distances Tie

If a point has equal squared distance to two centroids, the assignment needs a stated tie policy. This lesson pins one exact tie.

highlighted = computed this step

One point, two centroids

Use one displayed point p=(1, 0) with c1=(0, 0) and c2=(2, 0).

p=(1,0),c1=(0,0),c2=(2,0)p=(1, 0),\quad c_1=(0, 0),\quad c_2=(2, 0)
Equal-distance assignment tieA stated policy handles equal squared distances.equal-distance assignment tieone point, two centroids, equal squared distancessourcecoordinated^2 to pointpolicy resultpoint p(1,0)c1(0,0)1chosenc2(2,0)1same distancepolicy: lowest centroid index -> c1not a convergence, optimality, best-k, broad-data, or randomness claim

Both squared distances are equal

The squared distance from p to c1 is 1. The squared distance from p to c2 is also 1.

d2(p,c1)=1,d2(p,c2)=1d^2(p,c_1)=1,\quad d^2(p,c_2)=1
Equal-distance assignment tieA stated policy handles equal squared distances.equal-distance assignment tieone point, two centroids, equal squared distancessourcecoordinated^2 to pointpolicy resultpoint p(1,0)c1(0,0)1chosenc2(2,0)1same distancepolicy: lowest centroid index -> c1not a convergence, optimality, best-k, broad-data, or randomness claim

State the tie policy

The displayed policy assigns equal distances to the lowest centroid index, so p goes to c1. This is an assignment policy, not a convergence, optimality, best-k, generalization, or randomness claim.

equal distances→c1\text{equal distances}\rightarrow c_1
Equal-distance assignment tieA stated policy handles equal squared distances.equal-distance assignment tieone point, two centroids, equal squared distancessourcecoordinated^2 to pointpolicy resultpoint p(1,0)c1(0,0)1chosenc2(2,0)1same distancepolicy: lowest centroid index -> c1not a convergence, optimality, best-k, broad-data, or randomness claim