diff --git a/days/day20.st b/days/day20.st
index 931cd0f..e861d02 100644
--- a/days/day20.st
+++ b/days/day20.st
@@ -1,11 +1,12 @@
 Object subclass: Racetrack [
-    | grid distToEnd start end trackLen allCheats |
+    | grid distToEnd start end trackLen allCheats tree |
     rows: r
     [ grid := Grid new rows: r.
       self initTrack ]
 
     initTrack
-    [ | pos dir step |
+    [ | pos dir step posnList |
+      "Find the start, end, and count the path tiles..."
       trackLen := 1.
       grid allPosnsDo: [
         :p | | c | c := grid at: p.
@@ -19,6 +20,7 @@ Object subclass: Racetrack [
                         height: grid height
                         initWith: [:p | nil].
 
+      "Trace the path, obtaining the distances to the end..."
       pos := start.
       dir := {Posn up. Posn down. Posn left. Posn right} findFirstElt: [
           :dir | self isEmpty: start + dir ].
@@ -31,34 +33,43 @@ Object subclass: Racetrack [
           step := step + 1.
       ].
       distToEnd at: end put: 0.
-    ]
+
+      "Build a k-d tree to answer range queries
+       over the {distToEnd . x . y} dimensions."
+      posnList := OrderedCollection new.
+      distToEnd allPosnsDo: [
+          :p | (distToEnd at: p) ifNotNil: [
+              :dist | posnList add: {dist. p x. p y}]].
+      tree := KDTree fromPoints: posnList. ]
 
     isEmpty: pos [ ^(grid at: pos) = $. ]
 
     countCheatsWithTime: time
     [ | total |
       total := 0.
-      "allCheats := Bag new."
       grid allPosnsDo: [
           :pos | total := total + (self countCheatsFrom: pos withTime: time) ].
-      "allCheats printNl."
       ^total ]
 
     countCheatsFrom: pos withTime: time
-    [ | dist |
+    [ | dist count |
       dist := distToEnd at: pos.
       dist ifNil: [^0].
-      "dist < 100 ifTrue: [^0]."
-      ^self rangeCount: [
+      dist < 102 ifTrue: [^0].
+      count := 0.
+      self rangeDo: [
           :dpos :ddist |
-          self validCheatFrom: pos dist: dist
-               to: dpos dist: ddist
-               withTime: time
-      ]
-            inManhDistance: time
-            from: pos
-            maxDist: dist - 100. ]
+          (self validCheatFrom: pos dist: dist
+                to: dpos dist: ddist
+                withTime: time) ifTrue: [
+              count := count + 1 ]]
+           inManhDistance: time
+           from: pos
+           maxDist: dist - 102.
+      ^count ]
 
+    "Answers whether POS -> DPOS is a cheat that indeed only takes
+     TIME ps, and saves more than 100 ps."
     validCheatFrom: pos dist: dist
     to: dpos dist: ddist
     withTime: time
@@ -67,15 +78,16 @@ Object subclass: Racetrack [
                (dpos y - pos y) abs.
       ctime > time ifTrue: [^false].
       saved := dist - ddist - ctime.
-      "saved > 0 ifTrue: [allCheats add: saved]."
       ^saved >= 100 ]
 
-    rangeCount: block
+    "Call BLOCK with DPOS and DDIST for each position within
+     DISTMH of POS, which is at most MSCORE away from the end.
+     BLOCK may be called with other positions."
+    OLDrangeDo: block
     inManhDistance: distmh
     from: pos
     maxDist: mscore
-    [ | ddist dpos count px py doit |
-      count := 0.
+    [ | ddist dpos px py |
       dpos := Posn new.
       px := pos x. py := pos y.
       distmh negated to: distmh do: [
@@ -85,9 +97,94 @@ Object subclass: Racetrack [
                   dpos x: px + dx; y: py + dy.
                   ddist := distToEnd at: dpos.
                   ddist ifNotNil: [
-                      (block value: dpos value: ddist)
-                          ifTrue: [ count := count + 1 ]]]]].
-      ^count ]
+                      block value: dpos value: ddist ]]]] ]
+
+    "Optimisation of the above that uses the k-d tree to answer range queries."
+    rangeDo: block
+    inManhDistance: distmh
+    from: pos
+    maxDist: mscore
+    [ | min max dpos |
+      min := {0. pos x - distmh. pos y - distmh}.
+      max := {mscore. pos x + distmh. pos y + distmh}.
+      dpos := Posn new.
+      tree from: min to: max do: [
+          :ddist :dstx :dsty |
+          dpos x: dstx y: dsty.
+          block value: dpos value: ddist ] asSpreader ]
+]
+
+Object subclass: KDTree [
+    from: min to: max do: block
+    [ self subclassResponsibility ]
+]
+KDTree class extend [
+    fromPoints: points
+    [ ^self fromShuffledPoints: (Random new shuffle: points asArray) ]
+
+    fromShuffledPoints: points
+    [ ^self fromShuffledPoints: points axis: 1 ]
+
+    leafMaxSize [ ^5 ]
+    fromShuffledPoints: points axis: ax
+    [ | med splits dims nextAx value |
+      points size <= self leafMaxSize ifTrue: [^KDLeaf new points: points asArray].
+      med := self guessMedian: points axis: ax.
+      dims := med size.
+      nextAx := ax \\ dims + 1.
+      value := med at: ax.
+      splits := self splitPoints: points axis: ax on: value.
+      ^KDInner new
+           left: (self fromShuffledPoints: (splits at: 1) axis: nextAx)
+           right: (self fromShuffledPoints: (splits at: 2) axis: nextAx)
+           axis: ax
+           value: value ]
+
+    splitPoints: points axis: ax on: val
+    [ | l r fairLeft |
+      l := OrderedCollection new.
+      r := OrderedCollection new.
+      fairLeft := true.
+      points do: [
+          :point | | v onLeft |
+          v := point at: ax.
+          onLeft := (v < val) or: [
+              v = val and: [fairLeft := fairLeft not. fairLeft]].
+          (onLeft ifTrue: [l] ifFalse: [r]) add: point ].
+      ^{l . r} ]
+
+    medianGuessSize [ ^5 ]
+    guessMedian: points axis: ax
+    [ | arr |
+      arr := 1 to: (points size min: self medianGuessSize)
+               collect: [:i | points at: i].
+      arr sort: [:l :r | (l at: ax) <= (r at: ax)].
+      ^arr at: arr size // 2 + 1 ]
+]
+
+KDTree subclass: KDLeaf [
+    | points |
+    points: n [points := n. "self class leafSize: points size"]
+    from: min to: max do: block [ points do: block ]
+
+    printOn: st [ st << 'lf' << points ]
+]
+KDLeaf class extend [
+    | leafSizes |
+    leafSizes [ ^leafSizes ifNil: [leafSizes := Bag new] ]
+    leafSize: sz [ self leafSizes add: sz ]
+]
+
+KDTree subclass: KDInner [
+    | left right axis value |
+    left: l right: r axis: ax value: v
+    [left:=l. right:=r. axis:=ax. value:=v]
+    from: min to: max do: block
+    [ (min at: axis) <= value ifTrue: [ left from: min to: max do: block ].
+      (max at: axis) >= value ifTrue: [ right from: min to: max do: block ] ]
+
+    printOn: st [ st << '{' << left << ' <= [' << axis << '] '
+                  << value << ' <= ' << right << '}' ]
 ]
 
 AOC input: [ Racetrack new rows: stdin toLines asArray ];
diff --git a/days/utils.st b/days/utils.st
index b3f9f3d..e9647d2 100644
--- a/days/utils.st
+++ b/days/utils.st
@@ -218,6 +218,7 @@ Object extend [ chain [ ^MessageChain new pipelineValue: self ] ]
 
 Object subclass: Posn [
     | x y | x [^x] y [^y] x: n [x:=n] y: n [y:=n]
+    x: nx y: ny [x := nx. y := ny]
 
     + o [ ^Posn x: x + o x y: y + o y ]
     += o [ self x: x + o x; y: y + o y. ]
@@ -384,3 +385,15 @@ Integer extend [
         ^{s0 . t0}
     ]
 ]
+
+Random extend [
+    shuffle: coll
+    [ 1 to: coll size do: [
+          :i | | j tmp |
+          j := self between: 1 and: coll size.
+          tmp := coll at: i.
+          coll at: i put: (coll at: j).
+          coll at: j put: tmp.
+      ].
+      ^coll ]
+]