doc(algorithms, intervals): insert interval solution write up

Author: BrianLusina

algorithms/intervals/insert_interval/README.md

@@ -22,3 +22,54 @@ Return the updated list of intervals.
 
 ![Example 1](./images/examples/insert_interval_example_1.png)
 ![Example 2](./images/examples/insert_interval_example_2.png)
+
+## Solution
+
+We first want to create a new list merged to store the merged intervals we will return at the end.
+
+This solution operates in 3 phases:
+1. Add all the intervals ending before newInterval starts to merged.
+2. Merge all overlapping intervals with newInterval and add that merged interval to merged.
+3. Add all the intervals starting after newInterval to merged.
+
+### Phase 1
+
+In this phase, we add all the intervals that end before newInterval starts to merged. This involves iterating through the
+intervals list until the current interval no longer ends before newInterval starts (i.e. intervals[i][1] >= newInterval[0]).
+
+![Solution 1](./images/solutions/insert_interval_solution_1.png)
+![Solution 2](./images/solutions/insert_interval_solution_2.png)
+
+### Phase 2
+
+In this phase, we merge all the intervals that overlap with newInterval together into a single interval by updating 
+newInterval to be the minimum start and maximum end of all the overlapping intervals. This involves iterating through 
+the intervals list until the current interval starts after newInterval ends (i.e. intervals[i][0] > newInterval[1]).
+When that condition is met, we add newInterval to merged and move onto phase 3.
+
+![Solution 3](./images/solutions/insert_interval_solution_3.png)
+![Solution 4](./images/solutions/insert_interval_solution_4.png)
+![Solution 5](./images/solutions/insert_interval_solution_5.png)
+![Solution 6](./images/solutions/insert_interval_solution_6.png)
+![Solution 7](./images/solutions/insert_interval_solution_7.png)
+
+### Phase 3
+
+Phase 3 involves adding all the intervals starting after newInterval to merged. This involves iterating through the 
+intervals list until the end of the list, and adding each interval to merged.
+
+After completing these 3 phases, we return merged as the final result.
+
+![Solution 8](./images/solutions/insert_interval_solution_8.png)
+![Solution 9](./images/solutions/insert_interval_solution_9.png)
+![Solution 10](./images/solutions/insert_interval_solution_10.png)
+
+### Complexity Analysis
+
+#### Time Complexity
+
+O(n) where n is the number of intervals. We iterate through all intervals once to merge them.
+
+#### Space Complexity
+
+O(n) where n is the number of intervals. We need space for the merged output array.