Kth par improvement (#81)

* better constant

* [auto-verifier] verify commit c8b7231

---------

Co-authored-by: GitHub <noreply@github.com>

Author: lrvideckis

.verify-helper/timestamps.remote.json

@@ -70,7 +70,7 @@
 "tests/library_checker_aizu_tests/handmade_tests/dsu_size.test.cpp": "2024-09-22 18:42:16 -0500",
 "tests/library_checker_aizu_tests/handmade_tests/edge_cd_small_trees.test.cpp": "2024-09-21 13:04:45 -0500",
 "tests/library_checker_aizu_tests/handmade_tests/fib_matrix_expo.test.cpp": "2024-09-23 11:20:46 -0500",
-"tests/library_checker_aizu_tests/handmade_tests/lca_ladder_forest.test.cpp": "2024-10-07 00:04:04 -0700",
+"tests/library_checker_aizu_tests/handmade_tests/lca_ladder_forest.test.cpp": "2024-10-16 11:17:12 -0700",
 "tests/library_checker_aizu_tests/handmade_tests/manacher.test.cpp": "2024-10-06 21:59:37 -0700",
 "tests/library_checker_aizu_tests/handmade_tests/merge_sort_trees.test.cpp": "2024-09-22 18:42:16 -0500",
 "tests/library_checker_aizu_tests/handmade_tests/mobius.test.cpp": "2024-09-19 12:57:45 -0700",
@@ -138,8 +138,8 @@
 "tests/library_checker_aizu_tests/trees/edge_cd_contour_range_update.test.cpp": "2024-09-22 18:42:16 -0500",
 "tests/library_checker_aizu_tests/trees/edge_cd_count_paths_per_length.test.cpp": "2024-09-21 13:04:45 -0500",
 "tests/library_checker_aizu_tests/trees/edge_cd_reroot_dp.test.cpp": "2024-09-21 13:04:45 -0500",
-"tests/library_checker_aizu_tests/trees/kth_node_on_path.test.cpp": "2024-10-07 00:04:04 -0700",
-"tests/library_checker_aizu_tests/trees/ladder_decomposition.test.cpp": "2024-09-27 10:44:49 -0500",
+"tests/library_checker_aizu_tests/trees/kth_node_on_path.test.cpp": "2024-10-16 11:17:12 -0700",
+"tests/library_checker_aizu_tests/trees/ladder_decomposition.test.cpp": "2024-10-16 11:17:12 -0700",
 "tests/library_checker_aizu_tests/trees/lca_all_methods_aizu.test.cpp": "2024-10-07 00:04:04 -0700",
 "tests/library_checker_aizu_tests/trees/lca_all_methods_lib_checker.test.cpp": "2024-10-07 00:04:04 -0700",
 "tests/library_checker_aizu_tests/trees/subtree_isomorphism.test.cpp": "2024-09-19 12:57:45 -0700"

library/trees/ladder_decomposition/linear_kth_par.hpp

@@ -5,21 +5,22 @@
 //!   int kth_par = kp.kth_par(v, k);
 //! @endcode
 //! kth_par = a node k edges up from v
-//! @time O(3.5*n + q)
-//! @space O(3.5*n)
+//! @time O(n + q)
+//! @space O(n)
 struct linear_kth_par {
   struct node {
     int d, p = -1, idx, dl;
     vi lad;
   };
   vector<node> t;
-  vi j;
+  vector<pii> j;
   linear_kth_par(const vector<vi>& adj):
     t(sz(adj)), j(2 * sz(t)) {
     vi st;
     int pos = 1;
     auto add_j = [&]() -> void {
-      j[pos] = st[max(0, sz(st) - 1 - 2 * (pos & -pos))];
+      j[pos] = {st[max(0, sz(st) - 1 - 2 * (pos & -pos))],
+        st[max(0, sz(st) - 1 - 4 * (pos & -pos))]};
       pos++;
     };
     auto dfs = [&](auto&& self, int v) -> void {
@@ -37,26 +38,27 @@ struct linear_kth_par {
         }
       st.pop_back();
     };
-    rep(i, 0, sz(t)) if (t[i].p == -1) t[i].p = i,
-                                       dfs(dfs, i);
+    rep(i, 0, sz(t)) if (t[i].p == -1) dfs(dfs, i);
     rep(i, 0, sz(t)) if (
-      t[i].p == i || t[t[i].p].dl != t[i].dl) {
+      t[i].p == -1 || t[t[i].p].dl != t[i].dl) {
       int leaf = t[i].dl;
       vi& lad = t[leaf].lad;
       lad.resize(
-        min((t[leaf].d - t[i].d) * 7 / 2, t[leaf].d + 1),
+        min((t[leaf].d - t[i].d) * 2, t[leaf].d + 1),
         leaf);
       rep(k, 1, sz(lad)) lad[k] = t[lad[k - 1]].p;
     }
   }
   int kth_par(int v, int k) {
+    assert(0 <= k && k <= t[v].d);
     switch (k) {
     case 0: return v;
     case 1: return t[v].p;
     case 2: return t[t[v].p].p;
     default:
-      int i = 1 << __lg(k / 3),
-          leaf = t[j[(t[v].idx & -i) | i]].dl;
+      int i = 1 << __lg(k / 3);
+      auto [j1, j2] = j[(t[v].idx & -i) | i];
+      int leaf = t[t[v].d - t[j2].d <= k ? j2 : j1].dl;
       return t[leaf].lad[k + t[leaf].d - t[v].d];
     }
   }