diff --git a/src/diff.mo b/src/diff.mo
new file mode 100644
index 0000000..6cf7009
--- /dev/null
+++ b/src/diff.mo
@@ -0,0 +1,49 @@
+import Array "mo:stdlib/array";
+import Hash "mo:stdlib/hash";
+import HashMap "mo:stdlib/hashMap";
+import PQ "pq";
+
+module {
+    public type Time = Nat;
+    public type Diff = Int;
+    public type ValueDiff = {
+        time: Time;
+        diff: Diff;
+    };
+    public type Trace<Key> = HashMap.HashMap<Key, PQ.PQ<ValueDiff>>;
+    public type ExportTrace<Key> = [(Key, [ValueDiff])];
+
+    func ord(x: ValueDiff, y: ValueDiff): Bool { x.time < y.time };
+
+    public class Collection<K>(
+      keyEq: (K,K) -> Bool,
+      keyHash: K -> Hash.Hash) {
+        public let trace: Trace<K> = HashMap.HashMap(0, keyEq, keyHash);
+        public func exportTrace(): ExportTrace<K> {
+            var res: ExportTrace<K> = [];
+            for ((k, diffs) in trace.iter()) {
+                let vec_diffs = diffs.heap.toArray();
+                res := Array.append<(K, [ValueDiff])>(res, [(k, vec_diffs)]);
+            };
+            return res;
+        };
+        public func insert(k: K, t: Time) {
+            let diff = { time = t; diff = 1 };
+            let diffs = switch (trace.get(k)) {
+            case null { let pq = PQ.PQ(ord); pq.add(diff); pq };
+            case (?diffs) { diffs.add(diff); diffs };
+            };
+            ignore trace.set(k, diffs);
+        };
+        public func map(f:K -> K): Collection<K> {
+            let res = Collection<K>(keyEq, keyHash);
+            for ((k,diffs) in trace.iter()) {
+                let new_k = f(k);
+                for (diff in diffs.heap.iter()) {
+                    res.insert(new_k, diff.time);
+                }
+            };
+            return res;
+        }
+    };
+}
diff --git a/src/main.mo b/src/main.mo
index 014bf49..693a6ca 100644
--- a/src/main.mo
+++ b/src/main.mo
@@ -1,9 +1,11 @@
 import A "mo:stdlib/array";
-import P "mo:stdlib/prelude";
-import List "mo:stdlib/list";
+import B "mo:stdlib/buf";
+import Diff "diff";
 import H "mo:stdlib/hashMap";
 import Hash "mo:stdlib/hash";
-import B "mo:stdlib/buf";
+import List "mo:stdlib/list";
+import P "mo:stdlib/prelude";
+import D "mo:stdlib/debug";
 
 let N = 600;
 
@@ -137,4 +139,13 @@ actor {
     public query func output() : async ([Object], Int, Int, Int) {
         (E.objects.toArray(), E.pos.x, E.pos.y, E.pos.dir)
     };
+
+    public func testDiff(): async Diff.ExportTrace<Text> {
+        let a = Diff.Collection<Text>(varEq, Hash.hashOfText);
+        a.insert("a",0); a.insert("b",0);
+        a.insert("a",1); a.insert("c",1);
+        D.print(debug_show(a.exportTrace()));
+        let b = a.map(func (x:Text): Text = x#"_" );
+        return b.exportTrace();
+    };
 };
diff --git a/src/pq.mo b/src/pq.mo
new file mode 100644
index 0000000..d20a37a
--- /dev/null
+++ b/src/pq.mo
@@ -0,0 +1,65 @@
+import Buf "mo:stdlib/buf";
+import D "mo:stdlib/debug";
+
+module {
+    public class PQ<T>(ord: (T, T) -> Bool) {
+        public let heap = Buf.Buf<T>(0);
+        public func add(x: T) {
+            let i = heap.len();
+            heap.add(x);
+            up_heap(i);
+        };
+        /*
+        public func pop(): ?T {
+            let n = heap.len();
+            if (n == 0) {
+                return null;
+            };
+            let x = h.get(0);
+            let y = h.get(n - 1);
+            h.set(0, y);
+            // h.remove_last();
+            down_heap(pq, 0);
+            ?x
+        };
+        func down_heap<T>(pq: PQ<T>, i: Nat) {
+            let h = pq.heap;
+            let ord = pq.order;
+            let n = h.len();
+            let x = h.get(i);
+            func down_heap(j: Nat) {
+                if (2 * j + 1 < n) {
+                    let l = 2 * j + 1;
+                    let r = 2 * j + 2;
+                    let k = if (r < n and not ord(h.get(l), h.get(r))) { r } else { l };
+                    let y = h.get(k);
+                    if (ord(x, y)) {
+                        h.set(j, x);
+                    } else {
+                        h.set(j, y);
+                        down_heap(k);
+                    }
+                } else if (j != i) {
+                    h.set(j, x);
+                }
+            };
+            down_heap(i)
+        };*/
+        func up_heap(i: Nat) {
+            let x = heap.get(i);
+            func up_heap_(j: Nat) {
+                let k = (j - 1) / 2;
+                let y = heap.get(k);
+                if (j == 0 or ord(y,x)) {
+                    heap.set(j, x);
+                } else {
+                    heap.set(j, y);
+                    up_heap_(k);
+                }
+            };
+            if (i > 0) {
+                up_heap_(i);
+            };
+        };
+    };
+}