Sorting elements in a table with respect to another table

Question

I have two tables, each having three rows (R1, R2 & R3). The elements of R2 & R3 are same in two tables but in different orders. The elements in row R1 are different in two tables. I want to sort the elements of second table such that rows R2 & R3 become similar. For example, If

T1 = {{t1, 1, 2}, {t2, 2, 3}, {t3, 3, 2}, {t4, 2, 1}}

T2 = {{u1, 3, 2}, {u2, 1, 2}, {u3, 2, 1}, {u4, 2, 3}}

How can I sort elements of T2 such that second and third elements of each row are same as T1. That is

T3 = {{u2, 1, 2}, {u4, 2, 3}, {u1, 3, 2}, {u3, 2, 1}}

thanks

Answer 1

There are multiple ways. One is to use Ordering.

Let p1 and p2 be the permutations that order T1 and T2 by their R2 & R3.

p1 = Ordering[T1[[All, 2 ;; 3]]]
(* {1, 4, 2, 3} *)

p2 = Ordering[T2[[All, 2 ;; 3]]]
(* {2, 3, 4, 1} *)

Let's invert p1:

invp1 = Ordering[p1]
(* {1, 3, 4, 2} *)

Then we sort T2, then transform that to the order seen in T1:

T2[[p2]][[invp1]]
(* {{u2, 1, 2}, {u4, 2, 3}, {u1, 3, 2}, {u3, 2, 1}} *)

---

Another way is to use Association

a1 = AssociationThread[
  T1[[All, 2 ;; 3]],
  T1[[All, 1]]
  ]
(* <|{1, 2} -> t1, {2, 3} -> t2, {3, 2} -> t3, {2, 1} -> t4|> *)

a2 = AssociationThread[
  T2[[All, 2 ;; 3]],
  T2[[All, 1]]
  ]
(* <|{3, 2} -> u1, {1, 2} -> u2, {2, 1} -> u3, {2, 3} -> u4|> *)

This assumes that there were no duplicates {R2, R3} pairs.

KeyTake[a2, Keys[a1]]
(* <|{1, 2} -> u2, {2, 3} -> u4, {3, 2} -> u1, {2, 1} -> u3|> *)

If you have Mathematica 10.0+ and no trouble with duplicate keys, I would use the association-based method, and I would keep the data structures as associations instead of tables.

Answer 2

In addition to @Szabolcs's method, a more straightforward approach might be SortBy:

SortBy[T2, Position[T1[[All, 2 ;; 3]], #[[2 ;; 3]]] &]
(* {{u2, 1, 2}, {u4, 2, 3}, {u1, 3, 2}, {u3, 2, 1}} *)

which gives the answer you want.