diff --git a/.idea/.idea.osu.Desktop/.idea/modules.xml b/.idea/.idea.osu.Desktop/.idea/modules.xml
index 366f172c30..fe63f5faf3 100644
--- a/.idea/.idea.osu.Desktop/.idea/modules.xml
+++ b/.idea/.idea.osu.Desktop/.idea/modules.xml
@@ -2,7 +2,6 @@
 <project version="4">
   <component name="ProjectModuleManager">
     <modules>
-      <module fileurl="file://$PROJECT_DIR$/.idea/.idea.osu.Desktop/.idea/.idea.osu.Desktop.iml" filepath="$PROJECT_DIR$/.idea/.idea.osu.Desktop/.idea/.idea.osu.Desktop.iml" />
       <module fileurl="file://$PROJECT_DIR$/.idea/.idea.osu.Desktop/riderModule.iml" filepath="$PROJECT_DIR$/.idea/.idea.osu.Desktop/riderModule.iml" />
     </modules>
   </component>
diff --git a/osu.Android.props b/osu.Android.props
index 493b1f5529..a2c97ead2f 100644
--- a/osu.Android.props
+++ b/osu.Android.props
@@ -52,6 +52,6 @@
   </ItemGroup>
   <ItemGroup>
     <PackageReference Include="ppy.osu.Game.Resources" Version="2020.622.1" />
-    <PackageReference Include="ppy.osu.Framework.Android" Version="2020.623.0" />
+    <PackageReference Include="ppy.osu.Framework.Android" Version="2020.701.0" />
   </ItemGroup>
 </Project>
diff --git a/osu.Game.Tests/NonVisual/CustomDataDirectoryTest.cs b/osu.Game.Tests/NonVisual/CustomDataDirectoryTest.cs
index f3d54d876a..8ea0e34214 100644
--- a/osu.Game.Tests/NonVisual/CustomDataDirectoryTest.cs
+++ b/osu.Game.Tests/NonVisual/CustomDataDirectoryTest.cs
@@ -127,6 +127,9 @@ namespace osu.Game.Tests.NonVisual
                     var osu = loadOsu(host);
                     var storage = osu.Dependencies.Get<Storage>();
 
+                    // Store the current storage's path. We'll need to refer to this for assertions in the original directory after the migration completes.
+                    string originalDirectory = storage.GetFullPath(".");
+
                     // ensure we perform a save
                     host.Dependencies.Get<FrameworkConfigManager>().Save();
 
@@ -145,25 +148,25 @@ namespace osu.Game.Tests.NonVisual
                     Assert.That(storage.GetFullPath("."), Is.EqualTo(customPath));
 
                     // ensure cache was not moved
-                    Assert.That(host.Storage.ExistsDirectory("cache"));
+                    Assert.That(Directory.Exists(Path.Combine(originalDirectory, "cache")));
 
                     // ensure nested cache was moved
-                    Assert.That(!host.Storage.ExistsDirectory(Path.Combine("test-nested", "cache")));
+                    Assert.That(!Directory.Exists(Path.Combine(originalDirectory, "test-nested", "cache")));
                     Assert.That(storage.ExistsDirectory(Path.Combine("test-nested", "cache")));
 
                     foreach (var file in OsuStorage.IGNORE_FILES)
                     {
-                        Assert.That(host.Storage.Exists(file), Is.True);
+                        Assert.That(File.Exists(Path.Combine(originalDirectory, file)));
                         Assert.That(storage.Exists(file), Is.False);
                     }
 
                     foreach (var dir in OsuStorage.IGNORE_DIRECTORIES)
                     {
-                        Assert.That(host.Storage.ExistsDirectory(dir), Is.True);
+                        Assert.That(Directory.Exists(Path.Combine(originalDirectory, dir)));
                         Assert.That(storage.ExistsDirectory(dir), Is.False);
                     }
 
-                    Assert.That(new StreamReader(host.Storage.GetStream("storage.ini")).ReadToEnd().Contains($"FullPath = {customPath}"));
+                    Assert.That(new StreamReader(Path.Combine(originalDirectory, "storage.ini")).ReadToEnd().Contains($"FullPath = {customPath}"));
                 }
                 finally
                 {
diff --git a/osu.Game.Tests/Visual/Menus/TestSceneIntroWelcome.cs b/osu.Game.Tests/Visual/Menus/TestSceneIntroWelcome.cs
index 1347bae2ad..8f20e38494 100644
--- a/osu.Game.Tests/Visual/Menus/TestSceneIntroWelcome.cs
+++ b/osu.Game.Tests/Visual/Menus/TestSceneIntroWelcome.cs
@@ -2,6 +2,7 @@
 // See the LICENCE file in the repository root for full licence text.
 
 using NUnit.Framework;
+using osu.Framework.Audio.Track;
 using osu.Framework.Screens;
 using osu.Game.Screens.Menu;
 
@@ -14,15 +15,11 @@ namespace osu.Game.Tests.Visual.Menus
 
         public TestSceneIntroWelcome()
         {
-            AddAssert("check if menu music loops", () =>
-            {
-                var menu = IntroStack?.CurrentScreen as MainMenu;
+            AddUntilStep("wait for load", () => getTrack() != null);
 
-                if (menu == null)
-                    return false;
-
-                return menu.Track.Looping;
-            });
+            AddAssert("check if menu music loops", () => getTrack().Looping);
         }
+
+        private Track getTrack() => (IntroStack?.CurrentScreen as MainMenu)?.Track;
     }
 }
diff --git a/osu.Game/IO/OsuStorage.cs b/osu.Game/IO/OsuStorage.cs
index 499bcb4063..f5ce1c0105 100644
--- a/osu.Game/IO/OsuStorage.cs
+++ b/osu.Game/IO/OsuStorage.cs
@@ -24,12 +24,12 @@ namespace osu.Game.IO
             "storage.ini"
         };
 
-        public OsuStorage(GameHost host)
-            : base(host.Storage, string.Empty)
+        public OsuStorage(GameHost host, Storage defaultStorage)
+            : base(defaultStorage, string.Empty)
         {
             this.host = host;
 
-            storageConfig = new StorageConfigManager(host.Storage);
+            storageConfig = new StorageConfigManager(defaultStorage);
 
             var customStoragePath = storageConfig.Get<string>(StorageConfig.FullPath);
 
diff --git a/osu.Game/OsuGameBase.cs b/osu.Game/OsuGameBase.cs
index 3e7311092e..c79f710151 100644
--- a/osu.Game/OsuGameBase.cs
+++ b/osu.Game/OsuGameBase.cs
@@ -312,11 +312,13 @@ namespace osu.Game
             base.SetHost(host);
 
             // may be non-null for certain tests
-            Storage ??= new OsuStorage(host);
+            Storage ??= host.Storage;
 
             LocalConfig ??= new OsuConfigManager(Storage);
         }
 
+        protected override Storage CreateStorage(GameHost host, Storage defaultStorage) => new OsuStorage(host, defaultStorage);
+
         private readonly List<ICanAcceptFiles> fileImporters = new List<ICanAcceptFiles>();
 
         public async Task Import(params string[] paths)
diff --git a/osu.Game/Rulesets/UI/RulesetInputManager.cs b/osu.Game/Rulesets/UI/RulesetInputManager.cs
index ba30fe28d5..f2ac61eaf4 100644
--- a/osu.Game/Rulesets/UI/RulesetInputManager.cs
+++ b/osu.Game/Rulesets/UI/RulesetInputManager.cs
@@ -18,9 +18,6 @@ using osu.Game.Input.Handlers;
 using osu.Game.Screens.Play;
 using osuTK.Input;
 using static osu.Game.Input.Handlers.ReplayInputHandler;
-using JoystickState = osu.Framework.Input.States.JoystickState;
-using KeyboardState = osu.Framework.Input.States.KeyboardState;
-using MouseState = osu.Framework.Input.States.MouseState;
 
 namespace osu.Game.Rulesets.UI
 {
@@ -42,11 +39,7 @@ namespace osu.Game.Rulesets.UI
             }
         }
 
-        protected override InputState CreateInitialState()
-        {
-            var state = base.CreateInitialState();
-            return new RulesetInputManagerInputState<T>(state.Mouse, state.Keyboard, state.Joystick);
-        }
+        protected override InputState CreateInitialState() => new RulesetInputManagerInputState<T>(base.CreateInitialState());
 
         protected readonly KeyBindingContainer<T> KeyBindingContainer;
 
@@ -203,8 +196,8 @@ namespace osu.Game.Rulesets.UI
     {
         public ReplayState<T> LastReplayState;
 
-        public RulesetInputManagerInputState(MouseState mouse = null, KeyboardState keyboard = null, JoystickState joystick = null)
-            : base(mouse, keyboard, joystick)
+        public RulesetInputManagerInputState(InputState state = null)
+            : base(state)
         {
         }
     }
diff --git a/osu.Game/Rulesets/UI/Scrolling/Algorithms/SequentialScrollAlgorithm.cs b/osu.Game/Rulesets/UI/Scrolling/Algorithms/SequentialScrollAlgorithm.cs
index 0052c877f6..a1f68d7201 100644
--- a/osu.Game/Rulesets/UI/Scrolling/Algorithms/SequentialScrollAlgorithm.cs
+++ b/osu.Game/Rulesets/UI/Scrolling/Algorithms/SequentialScrollAlgorithm.cs
@@ -3,21 +3,26 @@
 
 using System;
 using System.Collections.Generic;
+using System.Diagnostics;
+using JetBrains.Annotations;
 using osu.Game.Rulesets.Timing;
 
 namespace osu.Game.Rulesets.UI.Scrolling.Algorithms
 {
     public class SequentialScrollAlgorithm : IScrollAlgorithm
     {
-        private readonly Dictionary<double, double> positionCache;
+        private static readonly IComparer<PositionMapping> by_position_comparer = Comparer<PositionMapping>.Create((c1, c2) => c1.Position.CompareTo(c2.Position));
 
         private readonly IReadOnlyList<MultiplierControlPoint> controlPoints;
 
+        /// <summary>
+        /// Stores a mapping of time -> position for each control point.
+        /// </summary>
+        private readonly List<PositionMapping> positionMappings = new List<PositionMapping>();
+
         public SequentialScrollAlgorithm(IReadOnlyList<MultiplierControlPoint> controlPoints)
         {
             this.controlPoints = controlPoints;
-
-            positionCache = new Dictionary<double, double>();
         }
 
         public double GetDisplayStartTime(double originTime, float offset, double timeRange, float scrollLength)
@@ -27,55 +32,31 @@ namespace osu.Game.Rulesets.UI.Scrolling.Algorithms
 
         public float GetLength(double startTime, double endTime, double timeRange, float scrollLength)
         {
-            var objectLength = relativePositionAtCached(endTime, timeRange) - relativePositionAtCached(startTime, timeRange);
+            var objectLength = relativePositionAt(endTime, timeRange) - relativePositionAt(startTime, timeRange);
             return (float)(objectLength * scrollLength);
         }
 
         public float PositionAt(double time, double currentTime, double timeRange, float scrollLength)
         {
-            // Caching is not used here as currentTime is unlikely to have been previously cached
-            double timelinePosition = relativePositionAt(currentTime, timeRange);
-            return (float)((relativePositionAtCached(time, timeRange) - timelinePosition) * scrollLength);
+            double timelineLength = relativePositionAt(time, timeRange) - relativePositionAt(currentTime, timeRange);
+            return (float)(timelineLength * scrollLength);
         }
 
         public double TimeAt(float position, double currentTime, double timeRange, float scrollLength)
         {
-            // Convert the position to a length relative to time = 0
-            double length = position / scrollLength + relativePositionAt(currentTime, timeRange);
+            if (controlPoints.Count == 0)
+                return position * timeRange;
 
-            // We need to consider all timing points until the specified time and not just the currently-active one,
-            // since each timing point individually affects the positions of _all_ hitobjects after its start time
-            for (int i = 0; i < controlPoints.Count; i++)
-            {
-                var current = controlPoints[i];
-                var next = i < controlPoints.Count - 1 ? controlPoints[i + 1] : null;
+            // Find the position at the current time, and the given length.
+            double relativePosition = relativePositionAt(currentTime, timeRange) + position / scrollLength;
 
-                // Duration of the current control point
-                var currentDuration = (next?.StartTime ?? double.PositiveInfinity) - current.StartTime;
+            var positionMapping = findControlPointMapping(timeRange, new PositionMapping(0, null, relativePosition), by_position_comparer);
 
-                // Figure out the length of control point
-                var currentLength = currentDuration / timeRange * current.Multiplier;
-
-                if (currentLength > length)
-                {
-                    // The point is within this control point
-                    return current.StartTime + length * timeRange / current.Multiplier;
-                }
-
-                length -= currentLength;
-            }
-
-            return 0; // Should never occur
+            // Begin at the control point's time and add the remaining time to reach the given position.
+            return positionMapping.Time + (relativePosition - positionMapping.Position) * timeRange / positionMapping.ControlPoint.Multiplier;
         }
 
-        private double relativePositionAtCached(double time, double timeRange)
-        {
-            if (!positionCache.TryGetValue(time, out double existing))
-                positionCache[time] = existing = relativePositionAt(time, timeRange);
-            return existing;
-        }
-
-        public void Reset() => positionCache.Clear();
+        public void Reset() => positionMappings.Clear();
 
         /// <summary>
         /// Finds the position which corresponds to a point in time.
@@ -84,37 +65,100 @@ namespace osu.Game.Rulesets.UI.Scrolling.Algorithms
         /// <param name="time">The time to find the position at.</param>
         /// <param name="timeRange">The amount of time visualised by the scrolling area.</param>
         /// <returns>A positive value indicating the position at <paramref name="time"/>.</returns>
-        private double relativePositionAt(double time, double timeRange)
+        private double relativePositionAt(in double time, in double timeRange)
         {
             if (controlPoints.Count == 0)
                 return time / timeRange;
 
-            double length = 0;
+            var mapping = findControlPointMapping(timeRange, new PositionMapping(time));
 
-            // We need to consider all timing points until the specified time and not just the currently-active one,
-            // since each timing point individually affects the positions of _all_ hitobjects after its start time
-            for (int i = 0; i < controlPoints.Count; i++)
+            // Begin at the control point's position and add the remaining distance to reach the given time.
+            return mapping.Position + (time - mapping.Time) / timeRange * mapping.ControlPoint.Multiplier;
+        }
+
+        /// <summary>
+        /// Finds a <see cref="MultiplierControlPoint"/>'s <see cref="PositionMapping"/> that is relevant to a given <see cref="PositionMapping"/>.
+        /// </summary>
+        /// <remarks>
+        /// This is used to find the last <see cref="MultiplierControlPoint"/> occuring prior to a time value, or prior to a position value (if <see cref="by_position_comparer"/> is used).
+        /// </remarks>
+        /// <param name="timeRange">The time range.</param>
+        /// <param name="search">The <see cref="PositionMapping"/> to find the closest <see cref="PositionMapping"/> to.</param>
+        /// <param name="comparer">The comparison. If null, the default comparer is used (by time).</param>
+        /// <returns>The <see cref="MultiplierControlPoint"/>'s <see cref="PositionMapping"/> that is relevant for <paramref name="search"/>.</returns>
+        private PositionMapping findControlPointMapping(in double timeRange, in PositionMapping search, IComparer<PositionMapping> comparer = null)
+        {
+            generatePositionMappings(timeRange);
+
+            var mappingIndex = positionMappings.BinarySearch(search, comparer ?? Comparer<PositionMapping>.Default);
+
+            if (mappingIndex < 0)
             {
-                var current = controlPoints[i];
-                var next = i < controlPoints.Count - 1 ? controlPoints[i + 1] : null;
+                // If the search value isn't found, the _next_ control point is returned, but we actually want the _previous_ control point.
+                // In doing so, we must make sure to not underflow the position mapping list (i.e. always use the 0th control point for time < first_control_point_time).
+                mappingIndex = Math.Max(0, ~mappingIndex - 1);
 
-                // We don't need to consider any control points beyond the current time, since it will not yet
-                // affect any hitobjects
-                if (i > 0 && current.StartTime > time)
-                    continue;
-
-                // Duration of the current control point
-                var currentDuration = (next?.StartTime ?? double.PositiveInfinity) - current.StartTime;
-
-                // We want to consider the minimal amount of time that this control point has affected,
-                // which may be either its duration, or the amount of time that has passed within it
-                var durationInCurrent = Math.Min(currentDuration, time - current.StartTime);
-
-                // Figure out how much of the time range the duration represents, and adjust it by the speed multiplier
-                length += durationInCurrent / timeRange * current.Multiplier;
+                Debug.Assert(mappingIndex < positionMappings.Count);
             }
 
-            return length;
+            var mapping = positionMappings[mappingIndex];
+            Debug.Assert(mapping.ControlPoint != null);
+
+            return mapping;
+        }
+
+        /// <summary>
+        /// Generates the mapping of <see cref="MultiplierControlPoint"/> (and their respective start times) to their relative position from 0.
+        /// </summary>
+        /// <param name="timeRange">The time range.</param>
+        private void generatePositionMappings(in double timeRange)
+        {
+            if (positionMappings.Count > 0)
+                return;
+
+            if (controlPoints.Count == 0)
+                return;
+
+            positionMappings.Add(new PositionMapping(controlPoints[0].StartTime, controlPoints[0]));
+
+            for (int i = 0; i < controlPoints.Count - 1; i++)
+            {
+                var current = controlPoints[i];
+                var next = controlPoints[i + 1];
+
+                // Figure out how much of the time range the duration represents, and adjust it by the speed multiplier
+                float length = (float)((next.StartTime - current.StartTime) / timeRange * current.Multiplier);
+
+                positionMappings.Add(new PositionMapping(next.StartTime, next, positionMappings[^1].Position + length));
+            }
+        }
+
+        private readonly struct PositionMapping : IComparable<PositionMapping>
+        {
+            /// <summary>
+            /// The time corresponding to this position.
+            /// </summary>
+            public readonly double Time;
+
+            /// <summary>
+            /// The <see cref="MultiplierControlPoint"/> at <see cref="Time"/>.
+            /// </summary>
+            [CanBeNull]
+            public readonly MultiplierControlPoint ControlPoint;
+
+            /// <summary>
+            /// The relative position from 0 of <see cref="ControlPoint"/>.
+            /// </summary>
+            public readonly double Position;
+
+            public PositionMapping(double time, MultiplierControlPoint controlPoint = null, double position = default)
+            {
+                Time = time;
+                ControlPoint = controlPoint;
+                Position = position;
+            }
+
+            public int CompareTo(PositionMapping other) => Time.CompareTo(other.Time);
         }
     }
 }
diff --git a/osu.Game/Screens/Ranking/ResultsScreen.cs b/osu.Game/Screens/Ranking/ResultsScreen.cs
index 968b446df9..49ce07b708 100644
--- a/osu.Game/Screens/Ranking/ResultsScreen.cs
+++ b/osu.Game/Screens/Ranking/ResultsScreen.cs
@@ -79,6 +79,11 @@ namespace osu.Game.Screens.Ranking
                                 RelativeSizeAxes = Axes.Both,
                                 Children = new Drawable[]
                                 {
+                                    statisticsPanel = new StatisticsPanel
+                                    {
+                                        RelativeSizeAxes = Axes.Both,
+                                        Score = { BindTarget = SelectedScore }
+                                    },
                                     scorePanelList = new ScorePanelList
                                     {
                                         RelativeSizeAxes = Axes.Both,
@@ -89,11 +94,6 @@ namespace osu.Game.Screens.Ranking
                                     {
                                         RelativeSizeAxes = Axes.Both
                                     },
-                                    statisticsPanel = new StatisticsPanel
-                                    {
-                                        RelativeSizeAxes = Axes.Both,
-                                        Score = { BindTarget = SelectedScore }
-                                    },
                                 }
                             }
                         },
diff --git a/osu.Game/osu.Game.csproj b/osu.Game/osu.Game.csproj
index 5f326a361d..3ef53a2a53 100644
--- a/osu.Game/osu.Game.csproj
+++ b/osu.Game/osu.Game.csproj
@@ -24,7 +24,7 @@
     <PackageReference Include="Microsoft.EntityFrameworkCore.Sqlite" Version="2.2.6" />
     <PackageReference Include="Microsoft.EntityFrameworkCore.Sqlite.Core" Version="2.2.6" />
     <PackageReference Include="Newtonsoft.Json" Version="12.0.3" />
-    <PackageReference Include="ppy.osu.Framework" Version="2020.623.0" />
+    <PackageReference Include="ppy.osu.Framework" Version="2020.701.0" />
     <PackageReference Include="ppy.osu.Game.Resources" Version="2020.622.1" />
     <PackageReference Include="Sentry" Version="2.1.4" />
     <PackageReference Include="SharpCompress" Version="0.25.1" />
diff --git a/osu.iOS.props b/osu.iOS.props
index 72f09ee287..492bf89fab 100644
--- a/osu.iOS.props
+++ b/osu.iOS.props
@@ -70,7 +70,7 @@
     <Reference Include="System.Net.Http" />
   </ItemGroup>
   <ItemGroup Label="Package References">
-    <PackageReference Include="ppy.osu.Framework.iOS" Version="2020.623.0" />
+    <PackageReference Include="ppy.osu.Framework.iOS" Version="2020.701.0" />
     <PackageReference Include="ppy.osu.Game.Resources" Version="2020.622.1" />
   </ItemGroup>
   <!-- Xamarin.iOS does not automatically handle transitive dependencies from NuGet packages. -->
@@ -80,7 +80,7 @@
     <PackageReference Include="Microsoft.EntityFrameworkCore.Sqlite" Version="2.2.6" />
     <PackageReference Include="Microsoft.EntityFrameworkCore.Sqlite.Core" Version="2.2.6" />
     <PackageReference Include="Newtonsoft.Json" Version="12.0.3" />
-    <PackageReference Include="ppy.osu.Framework" Version="2020.623.0" />
+    <PackageReference Include="ppy.osu.Framework" Version="2020.701.0" />
     <PackageReference Include="SharpCompress" Version="0.25.1" />
     <PackageReference Include="NUnit" Version="3.12.0" />
     <PackageReference Include="SharpRaven" Version="2.4.0" />