The author of Tarikh-i-Alai, which provides the details of the initial years of Sultan Alauddin Khilji, was Amir Khusrau.

Option 1: Ziauddin Barani - Ziauddin Barani was a prominent historian and chronicler during the reign of Sultan Muhammad bin Tughlaq. He is known for his work "Tarikh-i-Firoz Shahi" which covers the history of the Delhi Sultanate.

Option 2: Shams Siraj Afif - Shams Siraj Afif was a Persian poet and historian from the 14th century but he is not known for writing "Tarikh-i-Alai."

Option 3: Amir Khusrau - The correct answer. Amir Khusrau was a well-known poet, scholar, historian, and musician who lived during the reign of multiple Sultans in the Delhi Sultanate. He wrote several works, including "Tarikh-i-Alai" which delves into the early years of Alauddin Khilji`s rule.

Option 4: Yahiya bin Ahmad - There is no known historical evidence or reference to Yahiya bin Ahmad being the author of "Tarikh-i-Alai."

In conclusion